numerical method for pde by finite difference

8/16/2019 Numerical method for PDE by finite difference

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Overview 1 PDE 1-2 PDE 2 ODE 3 FD 4 FD 5 FD 6 FV 7-8 FV 8-9 FV 10

Numerical Methods for PDE: Finite Differencesand Finites Volumes

B. Nkonga

JAD/INRIA

Lectures Références:

Roger Peyret (NICE ESSI : 89),Tim Warburton (Boston MIT : 03-05),

Pierre Charrier (Bordeaux Matmeca 96-08)

B. NkongaLectures Références: Roger Peyret (

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1 Finite Difference(FD) and Finite volume(FV) : Overview

2 Modelization and Simplified models of PDE.3 Scalar Advection-Diffusion Eqation.

4 Approximation of a Scalar 1D ODE.

5 FD for 1D scalar poisson equation (elliptic).

6 FD for 1D scalar difusion equation (parabolic).

7 FD for 1D scalar advection-diffusion equation.

8 Scalar Nonlinear Conservation law : 1D (hyperbolic).

9 FV for scalar nonlinear Conservation law : 1D

10 Multi-Dimensional extensions


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Plan

1 Finite Difference(FD) and Finite volume(FV) : Overview2 Modelization and Simplified models of PDE.

3 Scalar Advection-Diffusion Eqation.









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Introduction : Why we need computation ?

To enjoy and understand !


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O i 1 PDE 1 2 PDE 2 ODE 3 FD FD FD 6 FV 8 FV 8 9 FV 10


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To prevent some humanity disaster, at least we hope so !B. Nkonga

Lectures Références: Roger Peyret (/ 33

O i 1 PDE 1 2 PDE 2 ODE 3 FD 4 FD 5 FD 6 FV 7 8 FV 8 9 FV 10


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To prevent consequences of some natural disaster !B. Nkonga


Overview 1 PDE 1 2 PDE 2 ODE 3 FD 4 FD 5 FD 6 FV 7 8 FV 8 9 FV 10


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ITER Int. Project.LMJ & NIF : Laser/plasmas

To help meet mankind’s future energy needs : Fusion


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Are computations possibles ?

Yes, Because we haveFundamental Laws

Conservation of mass,

Momentum, Energy.Laws of the Thermodynamics :(1st, 2nd, 3th) Gauss, Ampère’s,Faraday Laws

Computers are efficient

Tera-FLOPS computersavailables 1012 FLoating pointOperations Per Second.

Numerical approximation strategiesFinite volume, Finite element, Finite difference, Particles In Cells,..., structured/unstructured mesh, Parallel programming (MPI),


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Balance “relations” or equations !

1 9 9 8

2

2S

v

1v1

S

P r e m i e r e c o t e d

e

B o r d e a u x

Conservation Law : for Mass

Mass fluctuation, in a controlvolume, is the sum of outgoingand incoming mass :

Balance Relation

m(t+dt) = m(t)+ρ1S 1v1

−ρ2S 2v2

Balance equation

∂ρ

∂t + ∇ · (ρvvv) = 0


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ρ (x+dx)

ρ (x−dx)

v(x+dx)

x−dx x+dx

x

v(x−dx)

From relations to equations

m(x, t + dt) = m(x, t) +S t+dtt

f (x − dx,s)ds

−S t+dtt

f (x + dx, s)ds

where the flux is defined by f (x, t) = ρ(x, t)v(x, t)B. Nkonga




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ρ (x+dx)

ρ (x−dx)

v(x+dx)

x−dx x+dx

x

v(x−dx)

From relations to equations

m(x, t + dt) = m(x, t) + Sdtf (x− dx,t)− Sdtf (x + dx, t)

m(x, .) = 2Sdxρ(x, .)

where the flux is defined by f (x, t) = ρ(x, t)v(x, t)


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ρ (x+dx)

ρ (x−dx)

v(x+dx)

x−dx x+dx

x

v(x−dx)

From relations to equations dx → 0 and dt → 0

ρ(x, t + dt)− ρ(x, t)dt

= −f (x + dx, t)− f (x − dx,t)2dx

=⇒ ∂ρ∂t

+ ∂f

∂x = 0 =⇒ ∂ρ

∂t +

∂ (ρv)

∂x = 0

where the flux is defined by f (x, t) = ρ(x, t)v(x, t)


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Balance equations : general 1D case

∂ω

∂t +

∂f (ω)

∂x = S

where S can be defined, for example, by the chemistry process,geometrical topology ...

ω(x, t) =

ρ

ρY

ρu

E

and f (ω) =

ρu

ρY u

ρu2 + p(E + p)u


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∂ω

∂t +

∂f (ω)

∂x = S

where

S can be defined, for example, by the chemistry process,

geometrical topology ...

ω(x, t) =

ρ

ρY

ρu

E

and f (ω) =

ρu

ρY u

ρu2 + p

(E + p)u


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0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

0

0

0

0

0

0

0

0

0

0

0

1

1

1

1

1

1

1

1

1

1

1

S 1

S 2v1

v2

∂ω

∂t +

∂f (ω)

∂x = S

where S can be defined, for example, by the chemistry process,geometrical topology ...

ω(x, t) =

ρ

ρY

ρu

E

and f (ω) =

ρu

ρY u

ρu2 + p(E + p)u

B. Nkonga




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Finite volume Scheme : 1D case

∂ω

∂t +

∂f (ω)

∂x = S

Mesh and Control volumes


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i

i−1/2 i+1/2

i+1i−1

vin

∂ω

∂t +

∂f (ω)

∂x = S

Balance relations on a Control volume

ωn+1i − ωnitn+1 − tn +

φmi+1

2

− φmi− 1

2

xi+12

− xi− 12

= S mi


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i

i−1/2 i+1/2

i+1i−1

vin

ωn+1i − ωnitn+1 − tn +

φmi+12− φmi− 1

2

xi+12

− xi− 12

= S mi

How to define the numericalflux φ f B. Nkonga




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i

i−1/2 i+1/2

i+1i−1

vin

ωn+1i − ωnitn+1

− tn

+φmi+1

2

− φmi− 1

2

xi+12 − xi− 12=

S mi

Centered scheme :

φi+12

=f i + f i+1

2

Accurate but Unstable !


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i

i−1/2 i+1/2

i+1i−1

vin

ωn+1i − ωnitn+1 − tn

+φmi+1

2

− φmi− 1

2

xi+12 − xi− 12=S mi

Upwind scheme ( in this case) :

φi+12

= f i and φi−12

= f i−1

Less accurate but Stable √


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i

i−1/2 i+1/2

i+1i−1

vin

ωn+1

i − ωn

i

tn+1 − tn +φm

i+1

2 −φm

i−1

2

xi+12

− xi− 12

= S mi

General rule :The numerical flux should be consistent with the physicsUpwind : Follows information traveling in thecorrect direction.


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n3

n1v1

i

n2

2v

v3

aiωn+1i − ωnitn+1 − tn = −

j∈ϑ(i)

Φi,j

φi,j is now the flux crossing an interface from the cell i to j.B. Nkonga




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Need properties for Numerical Scheme

φi,j = φ(ωi, ω j)Properties we want the numerical approximation to satisfy are :

Consistency, Stability, Convergence

Accuracy : have a better result with a given mesh.Positivity and maximum principles ρ ≥ 0, ||v|| ≤ c.Second thermodynamic law (entropy production) : Nonlinearcases.

Lax Theorem for conservative systems

Consistency + Stability = Convergence


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Low Mach interface flow with surface Tension

100110

100111

100112

100113

100114

100115

100116

100117

100118

100119

100120

0 0.005 0.01 0.015 0.02 0.025

’Simulation’

’Exact’T= 120 ms

Laplace Law Recovered : δp = σκ

Initial Interface

T= 210 ms T= 690 ms


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Water drop in air, gravity and Surface tension


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Water drop in air, gravity and Surface tension


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Dam Break : Shallow water and Multiphase flow


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Shallow water and Multiphase flow


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Explosion and propagation : Mesh follows the shock font.


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Laser / Plasmas Interactions for Nuclear Fusion


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Computation and reality : Aerodynamic flow


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High speed aerodynamics : Shock and condensation

Condensation gives a view

of the shock wave profileB. NkongaLectures Références: Roger Peyret (

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Sh k D d


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Shock waves interactions : Diamond structure

FluidBox


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2 Modelization and Simplified models of PDE.










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numerical method for pde by finite difference

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