1. AMS 691 Special Topics in Applied Mathematics Lecture 3
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.
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. Fluid Transport Single species Multiple 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

12. 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.

13. Navier-Stokes Equations for Compressible Fluids

14. Incompressible Navier-Stokes Equation (3D)

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

16. 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

17. 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

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

19. Proposed Plan on UQ/QMU
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

20. Proposed Plan on 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.