1. AMS 691 Special Topics in Applied Mathematics Lecture 4 James Glimm Department of Applied Mathematics and Statistics, Stony Brook University Brookhaven National Laboratory

2. 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, elliptic This 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.

3. Research Issues Many PDEs are Nonlinear Multiple equations combined (multiphysics) To be solved numerically Multiscale, meaning that many different length scales are coupled Because nonlinear, numerical solutions are needed Because multiscale, numerical solutions are difficult, and require large scale computations Because multiphysics, accuracy and stability of coupling is important

4. Hyperbolic Equations The wave equation is the basic example of a hyperbolic equation.

5. Nonlinear Hyperbolic Conservation Laws

6. Parabolic, Elliptic Equations Hyperbolic processes govern wave motion Parabolic 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 diffusion Some 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 are very different from those for hyperbolic Since we have two or three types of terms in a single equation, we need multiple solution methods. Multiscale (example): parabolic term has small coefficient, important in thin layers only.

7. Typical equation forms

8. Parabolic equations

9. 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.

10. Numerical Solution for Hyperbolic + Parabolic: Operator Splitting

11. Operator split methods Operator split methods are only first order accurate.

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

13. 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.

14. Split methods and time step control Time steps for hyperbolic equations governed by a Courant-Friedrichs-Levi (CFL) condition.

15. CFL 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 split equations 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

16. Parabolic CFL 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 implicit solvers, there is no stability time step restriction (no CFL). But still an accuracy time step restriction.

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

18. Implicit Parabolic and sparse linear solvers Very different numerical issues here 1.Often use 3 rd party softrware Petsi, Nalib 2. Different kinds of algorithms Iterative multigrid or GMRES Special problem dependent approximate inverse Direct solvers Good for small systems. Poor scaling for large systems

19. 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

20. 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.

21. Navier-Stokes Equations for Compressible Fluids

22. Incompressible Navier-Stokes Equation (3D)

23. Turbulent mixing for a jet in crossflow and plans for turbulent combustion simulations

24. 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

25. 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

26. Scramjet Project –Collaborated Work including Stanford PSAAP Center, Stony Brook University and University of Michigan

27. 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?

28. Verification Compare to analytic solutions Compare to manufactured solutions Substitute 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 to original manufactured solution Mesh refinement: convergence? At expected order of accuracy? Symmetries, conserved quantities preserved? Asymptotic analysis. Small amplitude growth laws.

29. Validation Always requires experimental data. Data for validation must be totally independent of any data used in the simulation (to fix some parameters for example). Use of data usually requires statistics, to assess quality of fits.

30. UQ/QMU uncertainty 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

31. 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.