TOWARD PARALLEL, ADAPTIVE MESH REFINEMENT FOR CHEMICALLY REACTING FLOW SIMULATIONS1

Adaptive numerical methods offer greater efficiency than traditional numerical methods by concentrating computational effort in regions of the problem domain where the solution is difficult to obtain. In this paper, we describe progress toward adding mesh refinement to MPSalsa, a computer program developed at Sandia National Laboratories to solve coupled three-dimensional fluid flow and detailed reaction chemistry systems for modeling chemically reacting flow on large-scale parallel computers. Data structures that support refinement and dynamic load-balancing are discussed. Results using uniform refinement with mesh sequencing to improve convergence to steady-state solutions are also presented. Adaptive mesh refinement has been used with great success for a variety of applications (1, 5). With adaptive refinement, elements are subdivided into smaller elements in regions of the problem domain where greater resolution is needed. In this way, computational effort is concentrated in regions where it is most needed, without wasting high-resolution computation in other regions. While adaptive mesh refinement has been widely used on serial computers, its use on parallel computers is complicated by the need for distributed data structures, dynamic load balancing, data migration, and maintenance of a distributed element database. In this paper, we describe our efforts toward implementing parallel mesh refinement in MPSalsa, an unstructured finite element computer program developed at Sandia National Laboratories to solve coupled three-dimensional fluid flow and detailed reaction chemistry systems for modeling chemically reacting flow on large-scale parallel computers (12, 13). In particular, we describe the data structure design to support refinement and dynamic load balancing, the octree data structure used to store refined meshes, and issues that arise due to the implementation of refinement in parallel. We have implemented uniform refinement of 2D and 3D meshes as a step toward adaptive refinement. This step is a useful one. Through mesh sequencing, uniform refinement can be used to accelerate convergence of the nonlinear solver and enable steady-state solution of problems too difficult to solve from trivial initial guesses. We present three examples of the benefits of mesh sequencing. 2. OVERVIEW OF MPSALSA MPSalsa computes the solution of the conservation equations for momentum, total mass, thermal energy, and individual gas and surface phase chemical species for low Mach number flows. These equations, shown in Table 1, form a complex set of coupled, nonlinear PDEs. The continuous problem is spatially approximated by a Petrov-Galerkin finite element method. For transient problems, this spatial approximation is coupled with first- and second-order dynamically controlled time-stepping methods. Necessary transport properties, diffusion coefficients, kinetic rate constants and diffusion velocities are obtained from the CHEMKIN (10) library. The resulting nonlinear system is solved by an inexact Newton method with back-tracking (14). In the inexact Newton method, nonlinear residual information is used to determine the accuracy to which the linear subproblems are solved. Back-tracking is a technique for further improving the robustness of the nonlinear solver. It shortens a Newton step as needed to ensure that the nonlinear residual has been reduced adequately before the step is accepted. The Aztec library (9) of parallel preconditioned Krylov techniques is used to solve the resulting linear equations. The parallel Krylov algorithms implemented in Aztec include conjugate gradient, conjugate gradient squared, generalized minimal residual (GMRES) and transpose-free quasi-minimal residual methods. The available preconditioners are row sum and block Jacobi scaling, block Jacobi preconditioning, Neumann series and least-squares polynomial methods, and many additive Schwarz domain decomposition preconditioners using various incomplete LU factorizations with variable overlap.