ams 691 special topics in applied mathematics lecture 4 james glimm department of applied...
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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.