uncertain grid method for numerical solution of pde

UNCERTAIN GRID METHODFOR NUMERICAL SOLUTION OF PDE

OLEG V. DIYANKOV∗

Abstract. The Uncertain Grid Method for PDE solution is presented. The method is a nat-ural generalization of meshless methods based on the ideas of the free points and smooth particleshydrodynamics methods. The Uncertain Grid Method is illustrated with numerical examples of 2Dgas dynamics equations and Poisson equation solution.

Key words. PDE Numerical methods, Gas Dynamics, Poisson Equation, Meshless methods

AMS subject classifications. 76M25, 65M99, 65N99

1. Introduction. There are several different approaches to the construction ofnumerical methods for PDE solution. The most popular approaches are grid-basedmethods, notably finite difference (FD) and finite element (FE), and meshless meth-ods, notably particle-in-cell (PIC) and free-Lagrange (FL). While each method hasits own advantages and disadvantages, none of them is able to accurately model someof the most difficult problems encountered in advanced scientific research.

This paper is devoted to the introduction of a new approach for the solutionof PDEs, the Uncertain Grid Method (UGM). UGM builds on the advantages ofmeshless methods (PIC and FL) allowing the modeling of large deformations, butovercomes the disadvantages of these methods related to the nonmonotonicity andloss of accuracy of the derived numerical solutions. To help frame a discussion of thecapabilities and benefits of UGM, let us begin with an overview of the traditionalapproaches to the solution of PDEs.

Finite difference and finite element methods are used both for stationary andmoving Euler grids. There are a lot of very accurate and robust techniques basedon FE and FD representation of the numerical solution but they all share certaindisadvantages:

• Modeling of flows with large deformations (e.g. gas dynamics, magnetic hy-drodynamics, elastic-plastic deformations) leads to a grid crash and requiresthe application of special algorithms for grid reconstruction. The use of thesealgorithms leads to a smoothing of the numerical solution and to difficultieswith the description of interfaces especially in 3D cases.

• Modeling of complicated interfaces, which are common in the modeling ofdifferent instabilities, is quite difficult with FD and FE methods. The usualsolution to this shortcoming lies in accounting for the concentrations of themixing components, see [1], [2]. These methods are very complicated and havetheir own drawbacks. First, it is not readily evident how to modify the moreor less complicated equations of state in a cell with a mixture. Second, theinterface movement is also a challenge although some techniques for trackingthe interface do exist.

In attempts to overcome these drawbacks the so-called meshless methods havebeen developed. One of the most widely used families of meshless techniques isparticle-in-cell (PIC), a method initially proposed by Harlow [3]. Today there are

∗Neurok Techsoft LLC, office 120, 19a Oktyabrsky ave., Troitsk, Moscow region, Russia, 142190([email protected])

1

2 O. V. Diyankov

numerous modifications of the PIC method, see [4], [5], [6]. A good review of PICmethods is given by Belotserkovsky and Davydov [7].

The main idea of these methods is the introduction of two grids: the Euler grid(usually rectangular) used for the evaluation of gradients, and the Lagrange grid ofparticles for tracking the properties of matter. These grids exchange data with eachother: the matter properties are interpolated from the particles to the Euler grid, andthe gradients are interpolated from the Euler grid to the particles.

The free points method (FPM) [6] is slightly different in that it only needs knowl-edge of the data in the neighboring points. The new values are calculated usingintegral presentation of the solution via values in the actual point vicinity.

A similar idea gave start to the development of a new class of PIC methods calledsmooth particles hydrodynamics (SPH) [8]. The main idea of these methods is basedon the concept that any particle is supposed to be spread around its center with some

kernel (for example, 1πn/2·hn · e−

(r−r0)2

h2 , where n is the dimension of the problem).In this case the differentiating of the corresponding particle value is reduced to thedifferentiating of the kernel. The details of these methods can be found in [9], [10].

The main problems that appear while using these PIC methods are linked withtwo serious drawbacks:

• The first drawback is the necessity of approximation of the integral for FPMand approximation of the smooth function with radial basis functions. Thisapproximation is provided with a given accuracy for an equidistant distribu-tion of the particles, but this type of distribution can’t be achieved in themodeling of real flows with large deformations.

• The second drawback is linked with the necessity of partial derivatives ap-proximation via approximation of the original functions. In other words, inproviding a good approximation of the function in norm C or L2 we can’t besure that the approximation of the derivatives is good.

There have been attempts at overcoming these drawbacks, mostly linked withdifferent variations of SPH methods [10], but they did not result in sufficiently robustmethods.

Another class of meshless methods called ”Free Lagrangian” [11], [12] is basedon the representation of the region by a set of Voronoi cells. Though this approachmay provide robust code for modeling large deformations [13], there are two maindrawbacks:

• There is no universal solution to the problem of grid construction for anarbitrary arcwise connected region G. While some approaches exist, they donot work in all circumstances.

• There is a problem maintaining consistency between the ”geometrical” vol-ume and the volume obtained from the corresponding equations for volumechange.

This second drawback should be explained in more detail. The equation forLagrange volume change, ∂V

∂t = V · ~∇·~u, determines the change of the specific volumedepending on the velocity field. The geometrical volume in FL methods is determinedfrom Voronoi dissection and differs from the ”gas dynamic” volume (V = M ·V whereM is a mass assigned to a specific Voronoi cell) being a consequence of the Lagrangepoint movement described by the equation d~r

dt = ~u. As can be seen, the first equationis a differential consequence of the second. The problem is that this consequence doesnot remain in differences. This inconsistency generates nonmonotonicity and maylead to spurious extrema.

Uncertain Grid Method 3

This paper will document the construction of a meshless method that avoids thedrawbacks listed above. To begin with, let us consider gas dynamics equations inLagrange form:

∂∂t

∫V

dV =∮

∂V

undS

∂∂t

∫V

ρ · ~udV = − ∮∂V

p · ~ndS

∂∂t

∫V

ρ · (ε + 12 · |u|2)dV = − ∮

∂V

p · undS

(1.1)

Let us suppose that we have an arbitrary distribution of points having finitevolumes and all other values needed for the solution of an applied problem. Thesepoints are distributed in a finite volume bounded by a given piecewise linear surface.We suppose that the volume is convex.

To approximate 1.1 we need to get the approximation of the integral over thevolume of each particle and the approximation of the boundary integrals in the right-hand sides. We seek the approximation in the form:

∫V

fdV |i ≈∑

j∈Ωi∪iAi,j · fj

∮∂V

fdS|i ≈∑

j∈Ωi

Bi,j · fi,j(1.2)

Here Ωi is a set of numbers of neighboring particles for the particle number i, andfi,j is the approximation of function f at the ”uncertain face” between the i-th andj-th particles.

The coefficients in 1.2 should satisfy specific relationships. So for each particlewe have something like a cloud of particles around it which is used to construct theapproximation. As we see below, the power of this cloud (equal to the number ofparticles in it) can vary without significantly decreasing the quality of the approxi-mation.

Considering the first equation in 1.2 one can easily find a good and natural ap-proximation, Ai,j = |Vi|δij , where δij is the Kronecker tensor. This approximationhas a second order that gives a uniform distribution of the particles and is the simplestone. Higher order approximations may be useful when high order difference schemesare under construction.

In describing this new method the term ”particle” is a useful description, but inreality what we have is a volume which does not have strict bounds and about whichwe know certain values that are necessary for modeling a specific applied problem.The unbounded nature of this volume is the basis for the name ”Uncertain GridMethod”.

In the remainder of this paper, Section 2 discusses the conditions on Bi,j for theapproximation of 1.1. Section 3 examines how to get the coefficients while satisfyingthe conditions from Section 2. Section 4 is devoted to a discussion of the approx-imation of gas dynamics equations and Section 5 to the approximation of Poissonequations.

2. The Approximation Relationships. The approximation of the partial dif-ferential equations is based on the approximation of a set of basic operators. Theapproximation of the following operators should be considered:

4 O. V. Diyankov

f(~rk) ≈ 1|Vk| ·

∫Vk

fd~r

~∇f(xk) ≈ 1|Vk| ·

∮∂Vk

f · ~ndS

~∇ · ~g(xk) ≈ 1|Vk| ·

∮∂Vk

~g · ~ndS

(2.1)

The approximation of the first operator in 2.1 has been discussed above. In thispaper the simplest approximation of the integral operator for the k-th particle volume,

1|Vk| ·

∫Vk

fdV ≈ fk, is used.

The approximation of the divergence operator (the third operator in 2.1) is anevident consequence of the approximation of the gradient operator (the second oper-ator in 2.1), so we will only be discussing methods of approximation of the gradientoperator.

We try to find the weights Wij such that the integral in the second equation in2.1 may be approximated in the form:

∮

∂Vi

f~ndS ≈∑

j∈Ωi

Wij12(fj + fi)

~rj − ~ri

|~rj − ~ri|(2.2)

The goal of our treatment in the current section is to find the relationships forWij which lead to the approximations in 2.2 with the given order. The standardprocedure for the approximation treatment is used. Substitute in 2.2 fi and fj bythe values of the approximated differentiable function f in the points ~ri and ~rj anddecompose fj = f(~rj) in the Taylor series in the vicinity of ~ri.

fj = fi +∇fi · (~rj − ~ri) +12(~rj − ~ri)T · ∇2fi · (~rj − ~ri) + O(|~rj − ~ri|3)(2.3)

Substituting 2.3 in 2.2 and setting nij = ~rj−~ri

|~rj−~ri| , one can get:

|Vi|· < ∇f >i ≈ ∑j∈Ωi

Wijfi~nij + 12

∑j∈Ωi

Wij |~rj − ~ri|~nij~nTij∇fi+

14

∑j∈Ωi

Wij |~rj − ~ri|2~nij(~nTij∇2fi~nij) + Vi ·O(d2

i )(2.4)

Here < ∇f >i is the approximation of the corresponding gradient in the i-th

particle, and di =

∑j∈Ωi

Wij |~rj−~ri|2

∑j∈Ωi

Wij |~rj−~ri| is the diameter of the corresponding particle. The

following relationships should be fulfilled to get the first order approximation:

∑j∈Ωi

Wij~nij = 0

12

∑j∈Ωi

Wij |~rj − ~ri|~nij · ~nTij = |Vi| · I(2.5)

Here I is the identity matrix.


Returning to the problem statement, we recall that initially the domain has beenapproximated by a set of particles. Each particle has coordinates and various physicalvalues (depending on an applied problem), but no volumes are initially assigned tothe particles. When using FE or FL methods, one can determine the correspondingvolumes from geometrical considerations. The situation for UGM is different. Wedo not have a specific piece of space linked with a specific particle but we do haveapproximation of surface integrals. Remembering the formula for the volume of aclosed region with measurable boundary:

|V | = 1K·∮

∂V

~r · ~ndS(2.6)

where K is a dimension of the space of unknowns, one can write an approximationfor |Vi| in an internal particle:

|Vi| ≈ 1K·

∑

j∈Ωi

Wij~nTij · ~rij(2.7)

Here ~rij = 12 (~rj + ~ri). Using the first relationship in 2.5, 2.7 becomes:

|Vi| ≈ 12K

·∑

j∈Ωi

Wij~nTij ·

(~rj − ~ri

)(2.8)

Substituting 2.8 in 2.5, one can easily get the system of KN + K2N uniformequations, determining coefficients Wij :

∑j∈Ωi

Wij~nij = 0

12

∑j∈Ωi

Wij

(|~rj − ~ri|~nij · ~nT

ij − 1K · ~nT

ij ·(

~rj − ~ri

)· I

)= 0

(2.9)

For a 2D case, two of the six equations in 2.9 are dependent on the others. Theindependent set of equations is:

∑j∈Ωi

Wij~nij = 0

14

∑j∈Ωi

Wij |~rj − ~ri|(

nxijn

xij − ny

ijnyij

)= 0

12

∑j∈Ωi

Wij |~rj − ~ri|nxijn

yij = 0

(2.10)

It is evident that the equations in 2.9 have a trivial zero solution, so we needadditional equations that give us a non-trivial, non-negative solution Wij . To getthem it is necessary to consider approximation of the boundary conditions.

Before discussion of the boundary condition approximation, let us discuss a veryimportant issue linked with the calculation of the weights Wij . We have four equationsfor Wij for each particle, so to get a solution of these equations we evidently needat least four particles in the neighborhood of the given particle. This means that at

6 O. V. Diyankov

least four Wij should be nonzero for the given i. Of course a larger number of non-zero weights gives us more flexibility in finding appropriate Wij , though it results inlarger particle diameter which leads to worse approximation and to a larger numberof computational operations.

To obtain an optimal solution we follow the process proposed in SPH methods,namely, the introduction of a special area of vicinity for each particle. In this caseonly particles from this area are considered to be the neighbors for a given particle.We consider a particle to be a neighbor for the given one if the distance between themis smaller than m average diameters: |~rj − ~ri| ≤ m

2 ·(dj + di

). For our example we

consider 2 ≤ m ≤ 4.

2.1. Boundary Conditions Approximation. As discussed earlier, the regionin our example is convex and its boundary is a piecewise linear surface. Let us considera particle that is close to the boundary. The surface integral approximation for thisparticle can be written as a sum of two terms:

∮

∂Vi

f~ndS ≈∑

j∈Ωi

Wij12(fj + fi)

~rj − ~ri

|~rj − ~ri| +∑

j∈Θi

Wijfj~nj(2.11)

Here Θi is a set of boundary elements which are located in the vicinity of the i-thparticle, and ~nj is an external unit normal vector to the j-th boundary element. Eachboundary element contributes value

∑j∈Ξi

Wji · fBi in the total surface integral along

the external boundary. Here Ξi is a set of numbers of particles located in the vicinityof i-th boundary element, and fB

i is the value of the corresponding function in thecenter of i-th boundary element. Based on the above formula, one should impose thecorresponding restriction on Wij :

∑

j∈Ξi

Wji = Si(2.12)

Here Si is an area of the i-th boundary element.It is necessary to define the meaning of being ”located in the vicinity of” as used

above. We consider that the i-th particle is in the vicinity of boundary element j if thedistance from the particle to the element is less than m particle diameters, (~rj−~ri)T ·~nj ≤ mdi, and the nearest point at the boundary segment, ~r

Bj

i = ~ri +(~nj · (~rj −~ri)

),

belongs to it.Returning to the discussion of approximation relationships, let us discuss Equa-

tion 2.11. Substituting 2.3 in 2.11 and assigning fj to the nearest point of theboundary segment, ~r

Bj

i , one can get the following relationships for the weights ofthe ”boundary” particles:

∑j∈Ωi

Wij~nij +∑

j∈Θi

Wij~nj = 0

12

∑j∈Ωi

Wij |~rj − ~ri|~nij · ~nTij +

∑j∈Θi

Wij ·(~nT

j · (~rj − ~ri)) · ~nj · ~nT

j = |Vi| · I(2.13)

Taking into account the equation for the volumes of the boundary particles:


|Vi| ≈ 12K

·∑

j∈Ωi

Wij~nTij ·

(~rj − ~ri

)+

1K

∑

j∈Θi

Wij

(~nT

j · (~rj − ~ri))

(2.14)

one gets:

∑j∈Ωi

Wij~nij +∑

j∈Θi

Wij~nj = 0

12

∑j∈Ωi

Wij |~rj − ~ri|(

~nij · ~nTij − 1

K · I)

+

∑j∈Θi

Wij

((~rj − ~ri)T~nj

)(

nj · ~nTj − 1

K · I)

= 0

(2.15)

Although we examine the 2D case below, the formulas for the 3D case can bederived in the same manner.

The system of equations in 2.9 has only four independent equations because of thesymmetry of tensors in the second equation and equivalency of the equations for tensorelements 1.1 and 2.2, therefore we can leave only the equations for tensor elements1.1 and 1.2. Let us reformulate the system in 2.15 to get the first four equations closeto the four equations of 2.9. The resulting system is:

∑j∈Ωi

Wij~nij +∑

j∈Θi

Wij~nj = 0

14

∑j∈Ωi

Wij |~rj − ~ri|(nx

ij · nxij − ny

ij · nyij

)+

12

∑j∈Θi

Wij

((~rj − ~ri)T · nj

)(nx

j nxj − ny

j nyj

)= 0

12

∑j∈Ωi

Wij |~rj − ~ri|nxij · ny

ij+∑

j∈Θi

Wij

((~rj − ~ri)T · nj

)nx

j · ~nyj = 0

(2.16)

One more equation for Wij appears from the analysis of the system in 2.13.Taking the sum of the left-hand side of the second equation of the system over internalparticles indices i one gets:

∑i

(12

∑j∈Ωi

Wij(~rj − ~ri) · ~nTij +

∑j∈Θi

Wij

(~nT

j · (~rj − ~ri))~nj~n

Tj

)=

∑i

(12

∑j∈Ωi

Wij(~rj − ~ri) · ~nTij +

∑j∈Θi

Wij~rBj

i ~nTj +

~ri

( ∑j∈Ωi

Wij~nTij +

∑j∈Θi

Wij~nTj

))=

∑i

(12

∑j∈Ωi

Wij(~rj + ~ri) · ~nTij +

∑j∈Θi

Wij~rBj

i ~nTj

)=

∑i

(12

∑j∈Ωi

(Wij −Wji)(~rj + ~ri) · ~nTij +

∑j∈Θi

Wij~rBj

i ~nTj

)=

∑i

∑j∈Θi

(Wij~r

Bj

i

)~nT

j =∑j∈B

( ∑i∈Ξj

Wij~rBj

i

)~nT

j

(2.17)

8 O. V. Diyankov

The sum over the right-hand sides gives:

∑

i

|Vi| = |VG|(2.18)

Here |VG| is the volume of the region G.Let us return now to the description of the region G where the approximation

takes place. We suppose that it has a piece-wise linear boundary. Each segment hasthe length Sj , its center ~rj and the unit external normal ~nj . Therefore, it is easy tosee that the following formulas arise:

∑j∈B

Sj · ~nj =∮

∂G

~ndS = 0∑j∈B

xj · nxj · Sj =

∮∂G

x · nxdS = |VG|∑j∈B

yj · nyj · Sj =

∮∂G

y · nydS = |VG|∑j∈B

xj · nyj · Sj =

∮∂G

x · nydS = − ∫G

∇× ~(x, 0)dV =∫G

∂x∂y dV = 0

∑j∈B

yj · nxj · Sj =

∮∂G

y · nxdS =∫G

∇× ~(0, y)dV =∫G

∂y∂xdV = 0

(2.19)

System 2.19 implies:

|VG| · I =∑

j∈BSj~rj · ~nT

j(2.20)

Combining 2.17, 2.18 and 2.20, we get one more set of equations for boundaryelements:

Sj · ~rj =∑

i∈Ξj

Wij~rBi ∀j ∈ B(2.21)

The resulting system for the weights, determining the first order approximationfor the gradients, is:

∑j∈Ωi

Wij~nij +∑

j∈Θi

Wij~nj = 0

14

∑j∈Ωi

Wij |~rj − ~ri|(nx

ij · nxij − ny

ij · nyij

)+

12

∑j∈Θi

Wij

((~rj − ~ri)T · nj

)(nx

j nxj − ny

j nyj

)= 0

12

∑j∈Ωi

Wij |~rj − ~ri|nxij · ny

ij +∑

j∈Θi

Wij

((~rj − ~ri)T · ~nj

)nx

j · nyj = 0

∑i∈Ξj

Wij = Sj

∑i∈Ξj

Wij~rBj

i = Sj · ~rj

(2.22)

In addition to approximation relationships there is one more important propertyof good approximators of differential operators - conservatism.


2.2. Conservatism. Conservatism is equivalent to Stoke’s theorem for the re-gion. That means that the difference integral of the function gradient over the wholeregion should be equal to the difference integral of the function over the region bound-ary, and the expression

N∑

i=1

|Vi| < ∇f >i=N∑

i=1

( ∑

j∈Ωi

Wij12(fj + fi) · ~rj − ~ri

|~rj − ~ri| +∑

j∈Θi

Wijfj~nj

)(2.23)

should depend only on the boundary values of function f .After some simple transformations, 2.23 can be written in the form:

N∑

i=1

|Vi| < ∇f >i=∑

(i,j)∈R(Wij −Wji)

12(~fj + ~fi) · ~rj − ~ri

|~rj − ~ri| +∑

j∈BSjfj~nj(2.24)

Here R is a set of pairs of neighbor particle numbers in which i < j, and B isa set of numbers of boundary elements. Therefore, the symmetry of Wij is enoughfor conservatism. That is why we consider W to be a symmetrical matrix. The mainconclusion of this section can be formulated as the following theorem:

Theorem 1. Formulas 2.1 and 2.11 approximate the gradient with first order ofmagnitude with respect to max

idi if the weight Wij in the formulas is symmetric and

satisfies the equations in 2.22.

3. The Linear System for the Weights. The goal of this section is to treatthe system of equations for the weights and to propose a method for its solution. Tobegin with we try to treat the properties of the system of equations in 2.22. Initially itis necessary to find the rank of the system, and to educe a subsystem of the maximumrank.

3.1. The Linear System Rank. To better solve the system of equations, oneneeds to exclude the equations which are a consequence of other equations of thelinear system. To understand whether System 2.22 has such equations or not, we sumup each left-hand side of the equations. The first pair of equations gives the following:

∑i

∑

j∈Ωi

Wij~nij +∑

j∈Θi

Wij~nj

=

∑(i,j)∈R

(Wij −Wji)~nij +∑j∈B

Sj~nj =∑j∈B

Sj~nj = 0(3.1)

The sum of the third and fourth equations gives us an identity as can be seen fromthe previous section. In additiona, we see that all four sets of equations in 2.22 aredependent, so one equation in each set should be excluded to have a non-degeneratedsystem. For example, the equations for particle number one may be excluded.

3.2. The Statement of Problem for Weights Determination. We nowhave a system of equations in 2.22 for i = 2, ..., N and two conditions for Wij :

Wij = Wji

Wij ≥ 0(3.2)

10 O. V. Diyankov

Before calculation of the weights we must determine the weights graph. To dothis we set all Wij = 0 when the i-th and j-th particles are not neighbors or the i-thparticle is not in the vicinity of the j-th boundary element.

Since Wij is a sparse matrix, we can enumerate the non-zero elements of thematrix Wij assigning the same number to the (ij)-th and (ji)-th elements using thefollowing algorithm:

JNu = 1;for i = 1 by 1 to N

for j = i + 1 by 1 to Nif (Wij 6= 0)

then Numbij = JNu;JNu = JNu + 1;

endif ;end for;

end for;

(3.3)

Enumeration algorithm 3.3 is implemented more effectively by application of itsinner cycle only to non-zero elements of W . As a result of 3.3, the sparse integer matrixNumbij is used for 1D enumeration and the linear system 2.22 may be rewritten withrespect to one-dimensional vector α in the form:

A · α = b(3.4)

Here αNumbij = Wij = Wji. This formulation gives us the symmetrical solutionWij of the original linear problem 2.22.

There is one more restriction to the system 3.4:

α ≥ 0(3.5)

As we can see, the solution of 3.4 with restrictions 3.5 looks like a linear program-ming (LP) problem for minimizing

∑i

ci · αi under restrictions 3.4 and 3.5 where ci

are positive coefficients. The only difference is that we do not need to get the ”exact”minimum. The approximation takes place for any ”interior” point, that is any pointsatisfying restrictions 3.4 and 3.5. The constant in approximation estimation willdepend on how close we are to the exact minimum. From a practical point of viewwe do not need to come very close to the solution of the LP problem. The choice ofcoefficients ci has been treated in [17]. It was shown that the best choice is:

cNumbij = |~ri − ~rj |2(3.6)

The theory of LP can be found for example in [15], [16]. Below, the algorithmused by the author is described. We use here the primal-dual logarithmic barriermethod which results in the following linear system:

AT · y + s = cA · α = b

Ds ·Dα · e = µ · e(3.7)


Here Ds = Diag(s), Dα = Diag(α) and e = (1, 1, ..., 1)T . The solution of 3.7streams to the optimal one when µ → 0. As noted above, we do not need a ”very”optimal solution, so we can assume µ is equal to a constant, i.e. 1, and solve thecorresponding nonlinear equations in 3.7.

To solve 3.7 it is reasonable to use Newton method for a consequence of 3.7:

AT · y + µ ·D−1α · e = cA · α = b

(3.8)

Let us suppose that the ν-th iteration is done, so we have yν and αν , now thevalues at the (ν + 1) iteration can be derived from the system:

AT · yν+1 − µ ·D−2αν · (αν+1 − αν) =

(c− µ ·D−1

αν · e)A · (αν+1 − αν) = b−A · αν(3.9)

After simple transformations, system 3.9 reduces to the system for y:

A ·D2αν ·AT · yν+1 = µ(b−A · αν) + A ·D2

αν ·(c− µ−1 ·D−1

αν · e)(3.10)

The solution to yν+1 of 3.10 is substituted in the first equation of 3.9 and vectorαν+1 is obtained.

4. Central Difference Scheme for Gas Dynamics. This section describesthe Uncertain Grid Central Difference scheme (UGCD) for gas dynamic equations.The scheme construction is based on ideas from the paper [14]. In this paper wedescribe only the first order variant of UGCD. The results are illustrated with 1D and2D numerical examples.

The equations of Lagrangian gas dynamics can be written in the form:

∂∂t

∫V

dV =∮

∂V

undS

∂∂t

∫V

ρ · ~udV = − ∮∂V

p · ~ndS

∂∂t

∫V

ρ · (ε + 12 · |u|2)dV = − ∮

∂V

p · undS

(4.1)

Let us suppose that the region is bounded either by a rigid wall (~u · ~n = 0) orpiston (given (~uT · ~n)). We use the first order formulation of the KT scheme. Thescheme can be written in the form:

12 O. V. Diyankov

~rn+1i = ~rn

i + τ · ~uni(

1ρ

)n+1

i=

(1ρ

)n

i+ τ · ∑

j∈Θi

Wij · ~uj · ~nj + τ · ∑j∈Υi

Wij · ~ui · ~nj

+ τ · ∑j∈Ωi

12 ·Wij ·

((~ui + ~uj

)· ~rn

j −~rni

|~rnj−~rn

i| + cj+ci

1/ρj+1/ρi·(

1ρj− 1

ρi

))

~un+1i = ~un

i − τMi

· ∑j∈Θi

Wij · pi · ~nj − τMi

· ∑j∈Υi

Wij · pj · ~nj

− 12

τMi

∑j∈Ωi

Wij ·((

pj + pi

)· ~rn

j −~rni

|~rnj−~rn

i| −

cj+ci

2 ·(ρn

j · ~unj − ρn

i ~uni

))

εn+1i = εn

i + τMi

· ∑j∈Θi

Wij · pi · ~uj · ~nj + τ · ∑j∈Υi

Wij · pj · ~ui · ~nj

+ τMi

· ∑j∈Ωi

12 ·Wij ·

((pi · ~ui + pi · ~uj

)· ~rn

j −~rni

|~rnj−~rn

i|

+ cj+ci

2 · (ρj · εj − ρi · εi) + ρj · u2j

2 − ρi · u2i

2

)

(4.2)

Here Θi is the set of indices of the boundary elements with the ”given velocity”boundary condition adjacent to the i-th particle, and Υi is the set of indices of theboundary elements with the ”given pressure” boundary condition adjacent to the i-thparticle.

We can see that the volume |Vi|, density ρi and mass Mi are linked, |Vi| = Mi

ρi,

and it looks reasonable to conserve this relationship in the difference form. To do sowe use the equations for Wij in the form 2.13 setting |Vi| as a given value.

To understand the quality of the numerical solution given by 4.2 the followingproblem was modeled. Initially the gas is concentrated in the region [0, 0.4]×[−6, 1.5].The boundary condition at y = 1.5 − t is given the velocity un = −1. At the otherboundaries the condition is a ”rigid wall” that is given the velocity un ≡ 0. Initially,the gas in region −6 ≤ y ≤ 0 has the properties ρ = 1.0, ε = 10−6, uy = 0.0 andγ = 5/3. Gas in the region 0 ≤ y ≤ 1.5 has the properties ρ = 4.0, ε = 0.5, uy = −1.0and γ = 5/3. The results are presented in Figures 4.1 - 4.4.

−6 −5 −4 −3 −2 −1 00

0.5

1

1.5

−6 −5 −4 −3 −2 −1 01

1.5

2

2.5

3

3.5

4

4.5

Fig. 4.1. Pressure. Time 2.09 Fig. 4.2. Density. Time 2.09

The dependencies of all variables are shown at each particle. Therefore, since thenumber of particles in the x-direction was initially equal to 20, each 1D interval is


−6 −5 −4 −3 −2 −1 0−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

−6 −5 −4 −3 −2 −1 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Fig. 4.3. Uy. Time 2.09 Fig. 4.4. Eps. Time 2.09

represented by 20 points in the figures. As we can see, there is some nonmonotonicityat the shock wave while the number of 1D intervals at the shock is about 3. To avoidsuch nonmonotonicity the author proposes to use entropy viscosity.

4.1. Entropy Viscosity. The equations in 4.1 are the equations of isentropicflow, so the equation for the entropy should be fulfilled as:

T · dS = dE + P · dV(4.3)

Here V is a specific volume: V = 1ρ . At the shock wave the entropy takes a jump.

This increase in entropy is the main reason for the nonmonotonicity. Based on thisit is reasonable to introduce the viscosity which smoothes the entropy discontinuity.The central scheme of the equation for entropy with such viscosity looks like:

Tni · (Sn+1

i − Sni ) = τ

2 ·∑

j∈Ωi

Wij · cnij ·

(T nj Sn

j

Vnj

− T ni Sn

i

Vni

)(4.4)

As we do not have an entropy variable, it is reasonable to distribute the entropyviscosity change between energy and volume (density). This can be done in thefollowing way. Let us add the equation for the pressure increment dP :

dP = Pε · dE + PV · dV(4.5)

to 4.3. Having dP and T · dS given, we get the following expressions for dV and dE:

dE = T ·dS−dP/(ρ·PV)1−P ·Pε/PV

dV = dP/PV−Pε/PV ·T ·dS1−P ·Pε/PV

(4.6)

The formulas in 4.6 give us an idea of entropy viscosity distribution:

Eentr = τ2 ·

∑j∈Ωi

Wij · cij

Vij· T ·dS−dP/(ρ·PV)

1−P ·Pε/PV

ij

Ventr = τ2 ·

∑j∈Ωi

Wij · cij

Vij· dP/PV−Pε/PV ·T ·dS

1−P ·Pε/PV

ij

(4.7)

14 O. V. Diyankov

Here TdSij = (Ej −Ej)−Pij · (Vj −Vi) and dPij = Pj −Pi. The upper indicesare omitted as all values are calculated at the same n-th time step. The results ofmodeling the first problem are presented in Figures 4.7 - 4.8.

−12 −11 −10 −9 −8 −7 −6 −5 −40

0.2

0.4

0.6

0.8

1

1.2

1.4

−12 −11 −10 −9 −8 −7 −6 −5 −41

1.5

2

2.5

3

3.5

4

Fig. 4.5. Pressure. Time 7.95 Fig. 4.6. Density. Time 7.95

−12 −11 −10 −9 −8 −7 −6 −5 −4−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

−12 −11 −10 −9 −8 −7 −6 −5 −40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Fig. 4.7. Uy. Time 7.95 Fig. 4.8. Eps. Time 7.95

As we can see in these figures, the entropy viscosity application leads to a smoothersolution conserving a rather sharp wavefront.

4.2. Correlation of Volume and Coordinates Change. One more issue thatappears while using the described method is linked to the noncorrelation of the twoequations for coordinates and specific volumes. The equation for specific volume is aconsequence of the equation for coordinates as the specific volume is determined by:

V(~ξ,t)

V(~ξ,0)= ∇ξ · ~r(4.8)

Here ~ξ is the vector of Lagrange coordinates. Looking at the two equations below:

∂V∂t = V ~∇ξ · ~u∂~r∂t = ~u

(4.9)


we see that by taking the divergence of the second equation, we get the first equation.As such, the viscosity added to the first equation in 4.2 should correlate to the viscosityin the second equation. To obtain such correlation, the equation for coordinates atthe time tn may be written in the form:

∂~rn+1/2i

∂t = ~uni + δ~un

i(4.10)

The term δ~uni relates to the viscosity. Presenting δ~un

i = ∇φn and substituting itin the first equation of 4.9, we get:

Vni ·∆φn

i = ∂Vni

∂t − Vni · ~∇ξ · ~un

i(4.11)

The boundary condition for 4.11 is written in the natural form:

φn|G = 0(4.12)

Equation 4.11 with boundry condition 4.12 is solved using a difference schemedescribed in the next section. After obtaining the solution, δ~un

i is calculated:

δ~uni = 1

|Vi| ·∑

j∈Ωi

Wij · φi+φj

2 · ~rj−~ri

|~rj−~ri| + 1|Vi| ·

∑j∈Θi

Wijφi · ~nj(4.13)

Then δ~uni is substituted in Equation 4.10 and the new coordinates are evaluated.

Remark 1. Equation 4.11 with boundary condition 4.12 has a one-dimensionalkernel. To get a correct solution to the problem, one may just set φ to a given constantat a given point, i.e. φ(~r1) = 1. This means that we solve the corresponding differenceequations for all particles besides the first one.

Remark 2. It is reasonable to ask why we need to use such a complicatedprocedure for coordinate viscosity calculation. One answer is that the usual practicefor Lagrange methods is the elimination of the equation for specific volumes andcalculation cells (or elements) volumes via coordinates. This procedure works well forthe methods with fixed vicinity (fixed topology of the difference stencil).

With regard to irregular methods (FL, PIC or SPH), we should consider the re-configurable stencil. In this case the correctness of the procedure of volume calculationusing the coordinates of the cell (or element) nodes has to be proven. Some methodsdo not eliminate the equation for volumes and directly calculate the specific volumeusing a difference scheme. In this case a conflict between differential and differencerestrictions may appear. That is, 4.8 is not fulfilled in difference form due to vis-cosity terms. To avoid this conflict the described procedure of introducing numericalviscosity in the equations for coordinates has been proposed.

4.3. Example Using 2D Richtmayer - Meshkov Instability Calculation.It is well known that Richtmayer - Meshkov instability (RMI) takes place when ashock wave passes an interface between liquids (gases) having different densities. TheRMI modeling in this paper is presented as a sample usage of the proposed algorithmsfor modeling flows with large deformations.

Initially light gas is located in the region cos(x) ≤ y ≤ 9 and heavy gas is locatedin the region −9 ≤ y ≤ cos(x). No gravity is taken into account. The boundary

16 O. V. Diyankov

condition at the upper bound is a ”piston” with normal velocity equal to -0.3. Theboundary condition at the other boundaries is a ”wall”. The shock wave passes theinterface several times leading to an increase in the initial perturbation. The initialmatter distribution is shown in Figure 4.9. Subsequent distributions at various timesare shown in Figures 4.10 - 4.11.

0

2

4

6

8

-2

-4

-6

-8

0 1 2 3

Fig. 4.9. Matter distri-bution, time 0.0

0 1 2 3

−8

−6

−4

−2

0

2


0 1 2 3−10

−8

−6

−4

−2

0

2


As we can see in the figures, the results are reasonable. It should be notedthat there was no manual intervention in the calculations which shows the strongrobustness of the method.

5. Difference Scheme for Poisson Equation. It this section the differencescheme for the Poisson equation for UGM is discussed. The use of the describedapproach allows for developing difference schemes for elliptic and parabolic (diffusion)equations.

We consider the Poisson equation with Dirichlet boundary conditions. The regionG, where the solution is being sought, is bounded by a closed, piece-wise linear curve.The equation has the form:

∆u = fu|∂G = u0

(5.1)

If u0 and ∂G are smooth enough, function u0 can be continued smoothly in G


and Equation 5.1 for u can be written for u = u − u0 where u0 is a continuation ofu0 in G:

∆u = fu|∂G = 0

(5.2)

Here f = f −∆u.Below we consider Equation 5.2 and approximate this equation using the Un-

certain Grid approach. We omit ”hats” in variables u and f below. The naturalapproximation of the Poisson equation can be written in the form:

∑j∈Ωi

Wijuj−ui

|~rj−~ri| +∑

j∈Θi

Wij < ∇nu >Bj

i −|Vi| · fi = 0(5.3)

Here < ∇nu >Bj

i is a second order approximation of the normal gradient of theunknown function u at the point ~r

Bj

i = ~ri +((~rj − ~ri)T~nj

)~nj , that is: < ∇nu >

Bj

i ≈∇nu(~rBj

i ) + O(|dBj

j |2) where dBj

j is a diameter of the auxiliary point ~rBj

i . For theNeumann boundary condition this approximation is natural as the normal gradientis given.

For the Dirichlet boundary condition at 5.2 the following technique is used. To geta second order approximation of the normal gradient, let us find the approximationin the following form:

< ∇nu >Bj

i ≈ 2uj − ui

|~rBj

i − ~ri|− < ∇u >T

i ·~nj(5.4)

The approximation of < ∇u >i we take in the same form as earlier:

|Vi| < ∇nu >i≈∑

j∈Ωi

Wij~nijuj + ui

2+

∑

j∈Θi

Wijuj~nj(5.5)

Substituting Taylor series for uj in 5.5 and applying 2.22 we get:

|Vi| < ∇nu >i≈ |Vi|∇u(~ri) +∑

j∈Ωi

Wij |~rj − ~ri|2(

~nTij∇2u(~ri) · ~nij

)~nij+

∑j∈Θi

Wij |~rBj

i − ~ri|2(

~nTj · ∇2u(~ri) · ~nj

)~nj + |Vi|O(d2

i )(5.6)

To get the second order approximation it is necessary to have the second bracketin 5.6 be equal to zero. This gives us four more equations for Wij in the particlelocated in the boundary vicinity:

∑j∈Ωi

Wij |~rj − ~ri|2(~nij)3x +∑

j∈Θi

Wij |~rBj

i − ~ri|2(~nj)3x = 0∑

j∈Ωi

Wij |~rj − ~ri|2(~nij)2x(~nij)y +∑

j∈Θi

Wij |~rBj

i − ~ri|2(~nj)2x(~nj)y = 0∑

j∈Ωi

Wij |~rj − ~ri|2(~nij)x(~nij)2y +∑

j∈Θi

Wij |~rBj

i − ~ri|2(~nj)x(~nj)2y = 0∑

j∈Ωi

Wij |~rj − ~ri|2(~nij)3y +∑

j∈Θi

Wij |~rBj

i − ~ri|2(~nj)3y = 0

(5.7)

18 O. V. Diyankov

Adding 5.7 to 2.22 we get the resulting system of equations for the weight of thegraph of vicinity Wij which gives us the second order approximation of the gradientat the boundary. Applying this approximation to 5.4 we get the second order ap-proximation of the normal gradient at the boundary point ~r

Bj

i thus getting a goodapproximation of the Dirichlet boundary condition for the Poisson equation.

Let us now treat the approximation and stability of the presented differencescheme 5.3, 5.4 and 5.5.

5.1. Approximation of the Poisson Equation. Let us consider 5.3 for aninternal point. Presenting u(~rj) for internal particles and ∇nu(~rBj

i ) at boundaryelements as Taylor series:

u(~rj) = ui + (∇u)Ti · (~rj − ~ri) + 1

2 (~rj − ~ri)T · (∇2u)i · (~rj − ~ri)+O(|~rj − ~ri|3)

∇nu(~rBj

i ) = (∇u)Ti · ~nj + ((~rj − ~ri)T · ~nj)2~nT

j · (∇2u)i · ~nj+O(|((~rj − ~ri)T · ~nj |2 + d2

i )

(5.8)

and substituting 5.8 in the left-hand side of 5.3, we get:

(∇u)Ti

( ∑j∈Ωi

Wij~nij +∑

j∈Θi

Wij~nj

)+ Tr

(∇2ui ·

12

∑j∈Ωi

Wij |~rj − ~ri|~nij~nTij+

∑j∈Θi

Wij((~rj − ~ri)T · ~nj)~nj~nTj

)− |Vi|fi + |Vi|O(di)

(5.9)

Here Tr(Q) is the trace of a tensor Q that is the sum of the elements at the maindiagonal of the tensor. After simple transformations we can reduce 5.9 to the form:

|Vi|Tr

(∇2ui · I

)− |Vi|fi + |Vi|O(di) = |Vi|

(∆ui − f + O(di)

)(5.10)

5.10 means that the Poisson equation is approximated by 5.3 with first order withrespect to the maximum diameter of the particles. To finalize the treatment of theapproximation we need to prove that 5.4 has second order of magnitude with respectto di. Decomposing u(~ri) and (∇u)T (~ri) in the vicinity of ~r

Bj

i we get:

2 ·(∇u(~rBj

i )T · ~nj + 12 |~r

Bj

i − ~ri|~nTj · ∇2u(~rBj

i ) · ~nj + O(|~rBj

i − ~ri|2))−

(∇u(~rBj

i )T · ~nj + |~rBj

i − ~ri|~nTj · ∇2u(~rBj

i ) · ~nj + O(|~rBj

i − ~ri|2))

=

∇u(~rBj

i )T · ~nj + O(|~rBj

i − ~ri|2)

(5.11)

This proves the second theorem:Theorem 2. The difference scheme for the Poisson equation 5.3 - 5.4 approximates

the original equation with first order with respect to the maximum diameter of theparticles.


5.2. Stability of the Difference Scheme. Let us suppose that the solution isfound in the unit square. The difference scheme for 5.3 can be written in the form:

Ri · ui −∑

j∈Ωi

Qij · uj = bi(5.12)

We need to treat coefficients Rj and Qij for the particles. The difference schemefor the values in the particles may be written in the following form:

∑j∈Ωi

Wijuj−ui

|~rj−~ri| +∑

j∈Ψi

Wij < ∇nu >Bj

i −|Vi| · fi+

∑j∈Θi

Wij

(2 uj−ui

|~rBji−~ri|

− 1|Vi|

∑k∈Ωi

Wik(~nTik · ~nj)uk−ui

2 +

∑k∈Θi

Wik(uk − ui)(~nTk · ~nj)

)= 0

(5.13)

Here Θi is the set of the numbers of the Dirichlet boundary elements in thevicinity of the i-th particle, and Ψi is the set of the numbers of the Neumann boundaryelements. Let us combine the terms in 5.13 referring to the differences uj − ui whereuj refers to an internal particle:

∑k∈Ωi

Wikuk−ui

|~rk−~ri| ·(

1− 12|Vi|

∑j∈Θi

Wij(~nTik · ~nj)|~rk − ~ri|

)(5.14)

Considering the last sum in the brackets, we can see that taking into account theconvexity G:

∑j∈Θi

Wij(~nTik · ~nj)|~rk − ~ri| =

∑j∈Θi

Wij(~rk − ~ri)T · ~nj ≤∑

j∈Θi

Wij(~rj − ~ri)T · ~nj ≤ |Vi|(5.15)

This means that the coefficient in 5.14 before the difference uj − ui is positive.Now let us calculate the coefficient at the difference between the values in the

particle and a boundary element:

∑j∈Θi

Wijuj−ui

|~rBji−~ri|

(2− 1

|Vi|∑

k∈Θi

Wik|~rBj

i − ~ri|(~nTk · ~nj)

)(5.16)

Let us consider the last sum in 5.16. After some self-evident transformations weget:

1|Vi|

∑k∈Θi

Wik|~rBj

i − ~ri|(~nTk · ~nj) = 1

|Vi|∑

k∈Θi

Wik(~rBj

i − ~ri)T~nk =

1|Vi|

∑k∈Θi

Wik(~rBki − ~ri)T~nk + 1

|Vi|∑

k∈Θi

Wik(~rBj

i − ~rBki )T~nk

(5.17)

As mentioned above in Section Two, a particle has non-zero weight with a bound-ary element only if ~r

Bj

i ∈ G. In this case the following holds true: (~rBj

i −~rBki )T~nk ≤ 0.

20 O. V. Diyankov

This means that the sum in 5.17 is less than 1 and the coefficients before the differenceuj − ui in sum 5.16 is positive.

The calculations above lead us to the following conclusion. The coefficients Ri

and Qij are nonnegative and satisfy the equation, Ri ≥∑

j∈Ωi

Qij , and there exists i

for which the last inequality becomes strict. This fact implies the following theorem.Theorem 3. Any solution of the system of equations in 5.3 is bounded: |ui| ≤

maxk|fk|.The proof of the theorem is obtained using the standard technique, see for example

[18]. The theorem means that the numerical solution is stable in C norm. Accordingto the Lax theorem, Theorems 2 and 3 imply the convergence of the difference schemefor the Poisson equation for the Uncertain Grid Method.

5.3. Numerical Example. To understand the real properties of the proposedmethod, we solved a sample problem that has an exact solution:

∆u = −2 · π2 sin πx · sin πy(5.18)

The solution is sought in the unit square with the uniform Dirichlet boundaryconditions. The solution has the form:

u = sin πx · sin πy(5.19)

Initially all particles have been distributed uniformly; 40 in the x-direction by 40in the y-direction. The numerical results for the problem are shown in comparisonwith the exact solution in Figures 5.1 and 5.2.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

y

Fig. 5.1. Solution of the problem with

40 by 40 particles at x = 0.24375

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

y

Fig. 5.2. Solution of the problem with

40 by 40 particles at x = 0.95625

The exact solution is presented by the solid lines in the Figures, and numericalsolution is presented by the squares. As can be seen from the figures, there is closeagreement between the numerical results and the exact solution.

To get the numerical order of accuracy of the difference scheme, a couple ofadditional calculations for different numbers of particles have been performed. Threemore numerical statements have been considered: 10x10 particles, 20x20 particles and


80x80 particles. The results of modeling have been compared to the numerical resultsusing L2 norm. The resulting error norms are presented in Table 5.1.

Diam 10x10 20x20 40x40 80x802 1.962e-2 4.028e-3 9.860e-4 1.790e-43 3.486e-2 9.203e-3 2.021e-3 4.328e-44 6.827e-2 1.499e-2 3.894e-3 7.264e-4

Table 5.1. Numerical errors for Poisson equation solution

As we can see, the proposed scheme has second order accuracy for a uniformdistribution of particles. Of course such distribution cannot be achieved in real mod-eling, so the question of how the scheme behaves for an arbitrary distribution is worthinvestigating. To answer this question the following approach was taken.

Starting with initial uniform particle distribution, the particles were moved chaot-ically. At each step of the process the Poisson equation with the same right-hand side5.18 and the same boundary conditions 5.2 was solved and the error calculated usingC norm. We can see in Figure 5.3 the distribution of particles after 20 random steps.The error dependency at each iteration step is shown in Figure 5.4. As we can see,the error growth is less than two times. It is mostly linked with the growth of themaximum diameter of the particles di, thus enlarging the approximation error.

Fig. 5.3. Distribution of particles in 20 steps Fig. 5.4. Error dependence on the iteration

One more observation is linked with the choice of the area of vicinity. The firstcolumn in Table 5.1 refers to the number of diameters m used for the preliminarydetermination of the vicinity graph. As can be seen, the dependence on this parameteris not very significant, however, the larger the search area is for the neighboringparticles, the larger the approximation error.

6. Conclusion. The Uncertain Grid Method for PDE approximation can beused successfully for the modeling of flows with large deformations, especially whenit is necessary to take into account many physical processes involving the solution ofparabolic and elliptic equations (i.e. MHD). It can be also used for the solution ofPDEs in complicated regions.

The theoretical treatment and initial numerical experiments with the methoddescribed in this paper allow the author to make the conclusion that the proposedmethod has good approximation properties. It also may be applied to the construction

22 O. V. Diyankov

of high order difference schemes just by expanding the set of equations for the weightsof the vicinity graph.

Of course all of the advantages that UGM has over the traditional FD and FEmethods do not come for free - the payment is the computational cost of the solution ofthe corresponding LP problem. If one has a problem which can easily be solved usingthe traditional FD or FE methods, those methods should be used. However, whenthe problems of modeling large deformations or complicated interfaces appear, UGMcan provide improved accuracy and robustness, and the increased computational costscan be offset by eliminating manual intervention in the process of calculations.

Acknowledgments. The author wishes to thank the International Science andTechnology Foundation, Moscow, Russia for their partial support of his work in theframe of ISTC project #2830.

The author also wishes to thank Igor Krasnogorov for his help in numerical testingof the algorithms presented in this paper.

And I am very grateful to Vic Batson, who helped me a lot in making the expla-nations of the ideas presented in the paper clear and more understandable.

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uncertain grid method for numerical solution of pde

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