AMS 691Special Topics in Applied

MathematicsLecture 4

James Glimm

Department of Applied Mathematics and Statistics,

Stony Brook University

Brookhaven National Laboratory

Partial Differential Equations (PDEs) and Laws of Physics

Many laws of physics are expressed in terms of partial differential equations

Many types and varieties of partial differential equations.

Often nonlinear. Usually to be solved numerically, with some insight from theory

We have looked at nonlinear hyperbolic conservation laws. One basic class of physical laws.

A broader classification: hyperbolic, parabolic, ellipticThis is not the entire universe of PDEs, but is representative of many.Most common PDEs from physics will be one of these or a combination

Combination: many problems are put together by combining subproblems.

Research Issues

Many PDEs areNonlinearMultiple equations combined (multiphysics)To be solved numericallyMultiscale, meaning that many different length scales are

coupled

Because nonlinear, numerical solutions are neededBecause multiscale, numerical solutions are difficult, and require

large scale computationsBecause multiphysics, accuracy and stability of coupling is important

Hyperbolic EquationsThe wave equation is the basic example of a hyperbolic equation.

2 0

Rewrite as a first order system

0

( ) 0

Differentiate first equation with respect to t,

second equation with respect to x, and add:

Get ( ) . Equation for isentopic gas dynamic

tt

t x

t x

tt xx

U c U

v u

u p v

v p v

2

2 2

s.

Linearize: - '( )

Solutions (1 space dimension): ( ), ( )

Substitute, get '' '' 0

General solution: specify Cauchy data , at 0. Find

( ) ( ) (= general solution) from Caucht

p v c v

U x ct U x ct

c U c U

U U t

f x ct g x ct

y data.

Reference: Smoller Shock Waves and Reaction Diffusion Equations

Chapter 3, 17

Nonlinear Hyperbolic Conservation Laws

2

2

Wave equation:

0 10

0

0 1 acoustic matrix

0

Nonlinear:

( ) 0

0; ( ) /

Linearize: independent of

t x

t

t

U Uc

Ac

U F U

U A U A F U U

A U

Parabolic, Elliptic EquationsHyperbolic processes govern wave motionParabolic equations govern diffusion processes,Elliptic equations govern time independent phenomena,

either hyperbolic or parabolic

Multiphysics: hyperbolic + parabolic (+ elliptic)Some processes are combined wave motion and diffusionSome processes may be time independent, while others are not.Time evolution so rapid that steady state (d/dt = 0) is good approximation

Mathematical theory and numerical methods for parabolic/elliptic arevery different from those for hyperbolic

Since we have two or three types of terms in a single equation, we needmultiple solution methods.

Multiscale (example): parabolic term has small coefficient, important in thin layers only.

Typical equation forms2

2Elliptic: = 0

Parabolic:

(heat or diffusion equation)

Physical diffusion processes:

Thermal, mass concentration, momentum

Thermal diffusion in the energy equation

Mass diffusion in a specie

i i

t

UU

x

U U

s concentration equation

Momentum diffusion (viscosity) in the momentum equation

Parabolic equationsFundamental solution: Gaussian and

erf (= indefinite integral of Gaussian) 2

2 /( )

( )

2( )

( , ) fundamental solution of heat equation2

2( , 0) ( ) Dirac delta function at

( , ) = constant on parabolas ( ) / .

Diffusive information propagates

x y ty

y y

y

tf x t e

ff

tf x t x x y

f x t x y t const

0

0 ( )

0

like

General solution, with initial data ( ) is

( ) ( ) ( , )

( ) ( , ) solution of parabolic Riemann problem

(data = 1, y > 0)

y

x

t

f x

f x f y f x t

efr x f y t dy

Fluid Transport

• The Euler equations neglect dissipative mechanisms

• Corrections to the Euler equations are given by the Navier Stokes equations

• These change order and type. The extra terms involve a second order spatial derivative (Laplacian). Thus the equations become parabolic. Discontinuities are removed, to be replaced by steep gradients. Equations are now parabolic, not hyperbolic. New types of solution algorithms may be needed.

Numerical Solution forHyperbolic + Parabolic:

Operator Splitting

( )

At every time step, solve two equations in succession:

( ) 0

First step with hyperbolic methods;

second step with parabolic methods

t

t

t

U F U U

U F U

U U

Operator split methods

Operator split methods are only first order accurate.

2 2

22 2

22 2

( ) 2

1 ... 1 ...2 2

1 ...2

1 ( ) / 2 ...2

( ) / 2...

tA tB

t A B

t te e tA A tB B

tt A B A B

tt A B A B t AB BA

e t AB BA

Strang Splitting

2 2 2 2 2 2/2 /2

2 2 22

22

1 ... 1 ... 1 ...2 4 2 2 4

1 ( ) ...2 2 2

1 ( ) ...2

tA tB tA tA t A t B tA t Ae e e tB

t A At A B B AB BA

tt A B A B

Strang splitting is second order accurate. Higher commutators give still higher accuracy

Numerical methods:split vs unsplit

Split to add distinct physical processes, with different types of equations, different types of solvers.

Split to simplify equations, gain speed.

Split algorithms apply to spatial directions also.

( ) 0

( ) 0

( ) 0

t x

t y

t z

U F U

U F U

U F U

Split methods and time step control

Time steps for hyperbolic equations governed by a Courant-Friedrichs-Levi (CFL) condition.

where is maximum wave speed of problem.

Consider conservation law

0

0; /

0

| | 1( )

t

t

t x

t c x c

U U

U A U A F U

U UA

t xtA

U Ux

tAc CFL

x

CFL

1max imum wave speed from At x

Distinct physics and equations can have very different time scales and time steps. Operator splitting allows multiple time steps for fast physical systems. In other words, the operator splitequations can take several short steps for one equation, a single longer step for another.

Parabolic CFL:

Parabolic equations may need very short time steps. Why? What to do? (A) Operator split and small time steps (B) Implicit

2t c x

Parabolic CFL

2 2

2

; for example

/

As before, we need:

/ ; The constant depends on the elliptic operator A

t

t x

U AU A

U t x U

t x const

Forces small time steps. In a multiphysics problem with operator splitting, forces small time steps in the parabolic update only.

However, most parabolic solvers are implicit, not explicit. For implicitsolvers, there is no stability time step restriction (no CFL). But still anaccuracy time step restriction.

Implicit Parabolic Solvers

1 2 12

2

2 12

11 2

2

/

second difference operator in space

1 /

1 /

n n n

n n

n n

U U t x U

t x U U

U t x U

Parabolic and elliptic problems depend on inversion of a large sparse matrix. Usually very different in methods from those used for hyperbolic problems.

Implicit Parabolic and sparse linear solvers

Very different numerical issues here

1. Often use 3rd party softrware Petsi, Nalib 2. Different kinds of algorithms

Iterative multigrid or GMRESSpecial problem dependent approximate inverse

Direct solversGood for small systems. Poor scaling for large systems

Fluid Transport

• Single species– Viscosity = rate of diffusion of momentum

• Driven to momentum or velocity gradients

– Thermal conductivity = rate of diffusion of temperature• Driven by temperature gradients: Fourier’s law

• Multiple species– Mass diffusion = rate of diffusion of a single species in

a mixture• Driven by concentration gradients• Exact theory is very complicated. We consider a simple

approximation: Fickean diffusion

Comments

• Why study the Euler equations if the Navier-Stokes equations are more exact (better)?– Often too expensive to solve the Navier-Stokes

equations numerically– Often the Euler equations are “nearly” right, in that

often the transport coefficients are small, so that the Euler equations provide a useful intellectual framework

– Often the numerical methods have a hybrid character, part reflecting the needs of the hyperbolic terms and part reflecting the needs of the parabolic part.

Navier-Stokes Equationsfor Compressible Fluids

( )

is a (2+D)x(2+D) block diagonal matrix,

entries only in the momentum and

energy equations.

For multiple species, also entries for

each species concentration equation.

tU F U U

Incompressible Navier-Stokes Equation (3D)

( )tv v v P v

Turbulent mixing for a jet in crossflow and plans

for turbulent combustion simulations

The Team/Collaborators

• Stony Brook University– James Glimm– Xiaolin Li– Xiangmin Jiao– Yan Yu– Ryan Kaufman– Ying Xu– Vinay Mahadeo– Hao Zhang– Hyunkyung Lim

• College of St. Elizabeth– Srabasti Dutta

• Los Alamos National Laboratory– David H. Sharp– John Grove– Bradley Plohr– Wurigen Bo– Baolian Cheng

Outline of Presentation

• Problem specification and dimensional analysis– Experimental configuration– HyShot II configuration

• Plans for combustion simulations– Fine scale simulations for V&V purposes– HyShot II simulation plans

• Stanford simulation results

Scramjet Project

– Collaborated Work including Stanford PSAAP Center, Stony Brook University and University of Michigan

Some definitions

Verification: did you solve numerically (with controlled accuracy) the mathematical equations as posed?

Validation: does the totality of data (equations, boundary, initial conditions, equation parameters) reflect the reality of the physical problem being modeled (with controlled accuracy)?

Uncertainty quantification UQ): can you introduce error bounds for the V&V issues above?

Quantifiation of margins and uncertainties (QMU) what type of safety margins are needed to allow for all identified solution errors and uncertainties, to still assure correct performance of some engineered system?

Verification

Compare to analytic solutions

Compare to manufactured solutionsSubstitute a convenient function into the equation.It is not a solution, and leads to a nonzero right hand side.Regard this as a new equation, and solve it; compare tooriginal manufactured solution

Mesh refinement: convergence? At expected order of accuracy?

Symmetries, conserved quantities preserved?

Asymptotic analysis. Small amplitude growth laws.

Validation

Always requires experimental data.

Data for validation must be totally independent of any data used inthe simulation (to fix some parameters for example).

Use of data usually requires statistics, to assess quality of fits.

UQ/QMUuncertainty quantification.

Quantification of margins and uncertainties• Decompose the large complex system into several

subsystems

• UQ/QMU on subsystems

• Assemble UQ/QMU of subsystems to get the UQ/QMU for the full system

• Sub-system analysis goal: UQ/QMU for the essential subsystem --- combustor

UQ/QMU (continued)

• Our hypothesis is that an engineered system has a natural decomposition into subsystems, and the safe operation of the full system depends on a limited number of variables in the operation of the subsystems.

• For the scramjet, with its supersonic flow velocity, a natural time like decomposition is achieved, with each subsystem getting information from the previous one and giving it to the next.

• In this context, we hope that the number of variables to be specified at the boundaries between subsystems will be not too large. To show this in the scramjet context will be a research program, and central to the success of our objectives.

• We call the boundaries between the subsystems to be gates. Or rather the boundary and the specification of the criteria to be satisfied there is the gate.