parallel numerical simulation€¦ · derivation and analysis of models • of which type is the...

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Mathematical Models – ODE and PDEHans-Joachim Bungartz

Fifth SimLab Short Course on

Parallel Numerical Simulation

Belgrade, October 1-7, 2006

Mathematical Models – ODE and PDE

October 2, 2006

Hans-Joachim BungartzDepartment of Computer Science – Chair VTechnische Universität München, Germany

http://www5.in.tum.de/persons/bungartz.html

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1.1. Principles of Mathematical Modelling

• describe a given problem with some mathematical formalism inorder to

– get a formal and precise description– see fundamental properties due to the abstraction– allow a systematic treatment and, thus, solution

• (mathematical) model: formal description (and usually simplifi-cation) of (some) reality

• bigger or smaller evidence:

– exact natural science and engineering: long tradition (basicconservation laws of continuum mechanics, e.g.)

– economics, game theory, climate modelling: many openquestions (Keynes or not, MinMax or not, chaos or not?)

• to do: both derivation and analysis of models


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Derivation of a Model 1

• What do you want to model?

– a catalyst’s function or the detailed reactions in it?

• Which are the important quantities for that?

– Example 1: Optimum trajectory of the Space Shuttle – grav-itation of Pluto, gravitation of the Earth?

– Example 2: Prediction of Dow Jones Index tomorrow –statements of Mr. Greenspan, statements of myself?

• How important are they?

– Think of consequences of a neglection!

• What are their relations and interactions?

– qualitative and quantitative aspects

• How can these be (mathematically) described?

– algebraic or differential equations, graphs, automata,. . .


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Derivation of a Model 2

• description of relations and interactions:

– position, speed, and acceleration of an oscillating pendu-lum?

⇒ ordinary differential equation!

– deformation of a membrane under some load?⇒ partial differential equation!

– initial or boundary conditions of some growth process?⇒ algebraic equations!

– non-negativity of some quantity?⇒ algebraic inequality!

– order of several steps?⇒ graphs!

– state transitions?⇒ automata!


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Derivation and Analysis of Models

• Of which type is the resulting task?

– Is there a solution (Hamiltonian way in a graph)?– Find a/the solution (flow field around an aircraft)!– Find a/the best solution (shape optimization)!

• What can be said about solution(s) concerning

– their existence?– their uniqueness?– their dependency on the input data?

(well-posed problems: Hadamard 1923; Tikhonov, John; cf.however inverse problems)

• Is the model well-suited for a numerical treatment?

• Is the model derived so far correct?

– validation (experiments)!


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What to do with models?

• the analytical approach:

– prove existence and uniqueness formally– construct or find solution(s) formally/directly/analytically– desirable, but almost never possible

• the heuristic approach:

– trial and error, following some (hopefully smart) strategy– useful in discrete problems (travelling salesman etc.)

• the direct numerical approach:

– follow some numerical algorithm and end up with the exactsolution (Simplex algorithm for linear programming)

• the approximative numerical approach:

– approximate/discretize the model equations and end up withsome approximate solution


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1.2. Classes of Mathematical Models

• Models can be discrete or continuous:

– discrete models use a discrete/combinatoric description (in-teger numbers, graphs,. . . )

– continuous models use real quantities (real numbers, phys-ical quantities, differential equations,. . . )

– primarily, but not necessarily: discrete models for discretephenomena, continuous models for continuous phenomena

– examples: lattice-gas-automata for fluid flow, continuum me-chanics for traffic flow

• Models can be deterministic or stochastic:

– again no general relation between phenomena and models– roll the die: random phenomenon, stochastic model– crash test: deterministic phenomenon, deterministic model– but what about weather or data traffic through the internet?


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Discrete Models 1: Scheduling

• n jobs to be done on m machines working in parallel:

– no job simultaneously on more than one machine– no machine working simultaneously on more than one job

• several model parameters:

– characterizing the jobs: subjobs, processing time, earlieststarting time, due time, weighting, cost of delay, . . .

– characterizing machines and processing: identical or dif-ferent machines, order of subjobs important or not, prece-dence relations of jobs

– criteria of optimality: time of completion, delay, idle times

• task: find the best schedule with respect to some objective func-tion (minimizing overall time, e.g.)

• model: graph with disjunctive and conjunctive edges


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Discrete Models 2: Elections

• m candidates, n voters; model: theory of relations

• task: derive a reasonable collective preference from the differ-ent reasonable individual preferences

• possible choices:

– external dictator: individual preferences do not matter– internal dictator: one individual preference always wins– majority wins: cycles (a > b > c > a) cannot be excluded

– something else democratic — but what is democratic?

* all reasonable individual preferences are allowed

* the collective preference must be reasonable

* everything is possible with unanimity

* no dictator, independence of irrelevant alternativesimpossible if m > 2 and n > 1 (Arrow, 1951)!!

• drawback of democracy or drawback of our model??


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Discrete Models 3: Event Simulation

• task: model data or job traffic through computer system, opti-mize flow, find and avoid bottlenecks

• model: discrete event simulation, stochastic processes

– elementary queueing system (service stations, waiting pool)– quantities: waiting time, service time, staying time; filling,

capacity, throughput– arrival and completion of jobs: stochastic processes (deter-

ministic D, Markovian M, or general G)– notation: M|M|1 (arrival and completion follow a negative

exponential distribution, one station)– service strategies: FCFS, LCFS, random, round robin, . . .– queueing nets: network of elementary queueing systems

• computer systems: discrete space of states, state transitionsdriven by stochastic processes, Markovian chains or systems,resp.


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Hierarchy and Multiscale Property

• often to be chosen: scale/level of observation:

– Which resolution is necessary (w.r.t. the model’s accuracy)?– Which resolution can be tackled numerically?

• flow through a cylinder – how many dimensions?

1D: neglect cross-section2D: exploit symmetries3D: full resolution

• electric circuit simulation – spatial resolution or not?

– standard system simulators (SPICE, TITAN): only time, ODEs– parasitic cross effects: take space into account!

• turbulence – which vortices can be neglected?

– significant transport of energy between different scales– direct simulation – Large Eddy Simulation – averaging mod-

els


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Averaging and Homogenization

• often: coarse-grain phenomena are of interest, but fine-grainphenomena must not be neglected

• try to do some averaging:

– in time: turbulence, molecular dynamics– in space: flow and transport through porous media (a cata-

lyst or soil)

• formal concept: homogenization

– representative elementary volume– scaled reproduction, translation, periodic continuation– limit process of scaling factor– new quantities (effective parameters: porosity, permeabil-

ity)– new equations (porous media: instead of transport equa-

tions now Darcy-Forchheimer equation)


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1.3. Continuous Models 1: ODE

• in scientific computing: numerical simulations and, hence, typi-cally continuous models

• two big classes:

– problems with a treatment of space (involving partial differ-ential equations (PDE))

– problems without a treatment of space (involving ordinarydifferential equations (ODE))

• standard example for the latter: population dynamics

– development (growth) of populations,

* either isolated (without external influences)

* or in coexistence (peaceful or hostile) of different species– modelling has a long tradition– classical representative: model of Maltus (1798)


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Model of Maltus

• one species considered:

– constant birth rate γ per time unit and individual– constant death rate δ per time unit and individual– thus, constant growth rate λ = γ − δ

• development of p(t), the number of individuals:

p(t + ∆t) = p(t) + λp(t)∆t

(growth is proportional to size of population and time)

• this leads to the ODE

p(t) = λp(t)

with solutionp(t) = p0e

λt if p(0) = p0

• note:

– exponential growth or decrease– discrete reality, but continuous model!


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Model Refinement 1

• Is exponential growth realistic?

– population of the earth between 1700 and 1960: yes!

* growth rate of about 0.02, population doubles in 34.67years

– generally: no!

* limited capacity of the earth, limited resources

* increasing competiton for food, water, or air slows downgrowth

• refinement following Verhulst and others (19. century):

– population tends towards some saturation limit– linear birth and death rates (now per time unit only):

γ(t) = γ0 − γ1p(t) δ(t) = δ0 + δ1p(t) γ0 > δ0 > 0, γ1, δ1 > 0

big population decreases birth rate and increases death rate

– ODE: p(t) = γ(t)− δ(t) = −m · (p(t)− p∞)

– Limit exists if t tends towards infinity:

p(t) = p∞ + (p0 − p∞) · e−m·t


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Model Refinement 2

• Is Verhulst’s approach realistic?

– Second derivative of p(t) does not change its sign!– S-shape is widespread, however (cf. US population 1790-

1950, for example)

• further refinement in order to obtain S-shape:

p(t) = a · p(t)− b · p2(t), a > b > 0

with solution

p(t) =a · p0

b · p0 + (a− b · p0)e−at

• discussion:

– limit for t towards infinity is a/b

– S-shape for p0 < a/b

– example USA 1790-1950: a = 0.03134, b = 1.5587 · 10−10

– better than our starting point, but still no externalinfluences


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Model Refinement 3

• Typically, a is much bigger than b:

– The quadratic term gets influence only for really big p(t).– Why quadratic and not cubic? – That’s modelling!– justification: an individual is disturbed proportional to p(t)

• S-shape growth is called logistic

• For that, other ODEs can be used, too (widespread inmodelling of tumours growth, e.g.):

– p(t) = λ(t) · p(t) with some continuous, positive, anddecreasing function lambda (empirically, growth rates haveto decrease)

– or, even more general, p(t) = f(p(t)) · p(t) with somesuitable non-negative, decreasing, and vanishing (forincreasing t) f


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More than one Species

• next step of refinement: consider two species P and Q

p(t) = f(p(t), q(t)) · p(t),q(t) = g(p(t), q(t)) · q(t)

f and g defined for positive p and q

• A pair p, q with f(p, q) = g(p, q) = 0 allows for the definition of a

stationary solution(

p(t)q(t)

)=

(pq

)which is called equilibrium, if p > 0, q > 0.

• Is there an equilibrium? If yes, is it attractive?

– theory: a sufficient condition for attractiveness arenegative real parts of the eigenvalues of the Jacobian ofF (p, q) = (f(p, q)p, g(p, q)q) in p, q

– We study two special cases: predator-prey andcompetition characteristics.


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Competition Characteristics

• The species P and Q do not “eat” each other, but both strugglefor the same resources:

fp(p, q), fq(p, q), gp(p, q), gq(p, g) < 0 for p, q > 0

• Sufficient condition for attractive equilibrium reads

fp(p, q) · gq(p, q)− fq(p, q) · gp(p, q) > 0

(P’s influence on P is bigger than on Q and vice versa)

• simple concrete choice for f and g:

f(p, q) = a1 + a2 · p + a3 · q, g(p, q) = a4 + a5 · p + a6 · q

with (due to our model assumptions)

a1, a4 > 0, a2, a3, a5, a6 < 0, a2 · a6 > a3 · a5

• Attractive equilibrium f(p, q) = g(p, q) = 0, p, q > 0 exists!


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Predator-Prey Characteristics

• Q is “food” of P, which leads to a different growth behaviour:

fp(p, q), gp(p, q), gq(p, q) < 0, fq(p, q) > 0 for p, q > 0

a2, a5, a6 < 0, a3 > 0

(P, of course, enjoys an increasing population Q)

• Sufficient condition for attractive equilibrium is always fulfilled(i.e., if there is an equilibrium, it is attractive).

• classical representative: model of Volterra and Lotka:

a2 = a6 = 0

– no influence of P on P or Q on Q– There is an attractive equilibrium (though our sufficient

condition is not valid)!


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1.4. Continuous Models 2: PDE

• so far: only time as independent variable

• ODE-based population models sometimes too coarse:

– population in the USA during California gold rush in the1850s

– predictions of the UN concerning world population(industrialized countries versus third world)

• therefore: suppose p(x, t) or p(x, y, t) instead of p(t)

– California gold rush: 1D sufficient (east-west)– world population: perhaps 1D (north-south), perhaps 2D

• taking space into account makes models

– more accurate (spatial effects are no longer neglected)– more complicated (analytical solution becomes harder,

numerical solution means a lot of additional work)

• standard example: heat conduction


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Modelling with PDE

• taking space into account is typical for many problems orphenomena from physics or continuum mechanics:

– fluid mechanics: where will we get a tornado?– structural mechanics: where will be the crack?– process engineering: where is it how hot in the reactor?– electromagnetism: where is which electron density?– geology: where will the earthquake happen?

• more independent variables entail partial derivatives

• we distinguish:

– stationary problems: no time-dependence– unsteady problems: time-dependence (perhaps, but not

necessarily, with a stationary limit for increasing time)


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Heat Conduction

• central problem of thermodynamics

• let heat affect an object’s boundary – propagation?

– a wire, heated at one end– a metal plate, heated at one side– water cooling the reactor in a nuclear power plant– a room in winter: where to place the heating– a room in summer: effect of direct sunshine– boiling water in a pot on a ceramic hob

• central function of interest: temperature T

T (x, y) or T (x, y, t) or T (x, y, z, t)

• The values of T will depend on the material and its heatconductivity.


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Modelling Heat Conduction 1

• part 1 of the model: the PDE, indicating the relations ofchanges of T with respect to time and space (3D):

κ · (Txx + Tyy + Tzz) = κ ·(

∂2T

∂x2+

∂2T

∂y2+

∂2T

∂z2

)=

∂T

∂t= Tt

or shortly κ ·∆T = Tt with the Laplace operator ∆

• short derivation (excursion to physics):

– starting point is the basic principle of energy conservation– changes of heat in some part D of our domain are due to

flux in/out D’s surface and to external sources and drainsin D

∂

∂t

∫D

ρcTdV =

∫D

qdV +

∫∂D

k∇T · ~n dS

– density ρ, specific heat c, external term q, heat conductivityk, outer normal vector ~n, volume/surface element dV/dS


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• derivation of the heat equation (continued):

– transform the above equation according to Gauß’ theorem:∫D

(ρcTt − q − k∆T )dV = 0

– This holds for an arbitrary part D of our domain. Hence,the integrand must vanish:

Tt = κ∆T +q

ρc, κ =

k

ρc

– κ > 0 is called the thermal diffusion coefficient (since theLaplace operator stands for a (heat) diffusion process)

– For vanishing external influence q = 0, we get (and, thus,have derived) the famous heat equation:

Tt = κ∆T


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• part 2 of the model: the PDE needs boundary orinitial-boundary conditions to provide a unique solution:

– Dirichlet boundary conditions: fix T on (part of) theboundary

T (x, y, z) = ϕ(x, y, z)

– Neumann boundary conditions: fix T ’s normal derivativeon (part of) the boundary:

∂T

∂n(x, y, z) = ϕ(x, y, z)

– pure Dirichlet and mixtures are allowed, pure Neumannb.c. do not lead to a unique solution (with T solvesT + const. the PDE, too)

– in case of time-dependence: initial conditions for t = 0

• in case of no time-dependence: Laplace equation


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• meaning of boundary conditions:

– Dirichlet: the temperature T is prescribed itself along(part of) the boundary (some defined heating or cooling)

– Neumann: the temperature flux through (part of) theboundary is prescribed (if vanishing: complete isolation,no orthogonal transport of heat into or out of the domain

• analytical solutions:

– In simple (1D) configurations, solutions can be givenexplicitly via separation of variables (Fourier’s method).We will discuss these in the exercises.

– The heat equation is a simple case of a PDE, wheregeneral statements concerning existence and uniquenessof solutions are possible. Often, such theorems can not beproven.


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Types of PDE

• The heat equation is a linear PDE of second order:d∑

i,j=1

ai,j(~x) · uxi,xj(~x) +

d∑i=1

ai(~x) · uxi(~x) + a(~x) · u(~x) = f(~x)

• three types are distinguished:

– elliptic PDE: the matrix A of the ai,j is positive or negativedefinite

– parabolic PDE: one eigenvalue of A is zero, the othershave the same sign, and the rank of A together with thevector of the ai is full (d)

– hyperbolic PDE: A has 1 positive and d− 1 negativeeigenvalues or vice versa.

• examples:

– elliptic: Laplace equation ∆u = 0

– parabolic: heat equation ∆u = ut

– hyperbolic: wave equation ∆u = utt


parallel numerical simulation€¦ · derivation and analysis of models • of which type is the...

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