Robust approximation of nonlinear conservation equations
Abstract
Research objectives: We have recently established a rigorous research program for constructing numerical methods to solve nonlinear conservation equations. These methods are very robust and efficient up to second-order accuracy in space. They can be used to simulate nonlinear phenomena dominated by hyperbolicity where the formation of shocks, interfaces, vacuum, or dry states are important like in flooding, dam-breaking, supersonic flows, magnetohydrodynamics, combustion. Such phenomena arise in countless applications of interest to the Army. Our goal is to push forward this research program and to construct methods that are third and higher order accurate in space, are computationally efficient, and are robust at the same time. Robustness here means that the numerical solution is guaranteed to satisfy all the physical bounds of the exact solution. This is an extremely difficult task, and we are not aware of the existence in the community of any method with all of these properties. The objective is to construct a set of methods that are third or higher order accurate in space and at least as efficient as the second order method we have developed so far for non-smooth solutions (strong shock waves and contact waves). This new set of methods will work with arbitrary meshes in any space dimension and will be easily parallelizable. They will be applicable to a large class of problems in hydrodynamics, aerodynamics, multimaterial mechanics, combustion, magnetohydrodynamics, etc. Intellectual merit: Many numerical methods have some of the desired properties (spectral, finite volumes, discontinuous Galerkin). These methods are high-order accurate for smooth solutions, but their performance deteriorates when vacuum (or dry state) is present or strong shocks appear. In these cases the CFL number goes to zero and nonphysical oscillations appear, which makes these methods unusable in realistic applications. We have discovered over the last few years that, contrary to what was commonly believed, robustness does not depend on the underlying approximation space: continuous Galerkin, discontinuous Galerkin, or finite volumes. Actually, we have been able to construct second-order methods that are robust and are discretization agnostic (i.e., work with continuous finite elements, discontinuous finite elements, and finite volumes) even in the presence of source terms. Our methods satisfy all the physical requirements (maximum principle for scalar and invariant domain for systems, optimal CFL, optimal cost) and work on arbitrary meshes in any space dimension. We will construct a new set of methods that are at least third-order accurate in space (time accuracy is not an issue) by proceeding as follows: we will extend our convex limiting technique to blend a first or second order robust method (which will be used to enforce the invariant domain properties) with a high-order method (entropy-viscositybased). The key difficulty is to make the stencil of the low-order method as small as possible and make the low and the high order method conserve the same mass at the same time. Broader impact: The proposed research program will improve algorithms in computational fluid dynamics (Euler, multiphase flows, combustion, LES). The new method will be robust, have no tuneup constants, and work with any geometry. The methodology will be applicable in many discretization frameworks: continuous finite elements, finite volumes and discontinuous Galerkin. The proposed technique will be competitive in the race to exascale computing. It will have a broad impact in many fields in engineering. Proposing a novel technique for solving nonlinear hyperbolic problems with source terms which develop shocks or sharp interfaces will benefit every area of science and engineering where controlling or dealing with this type of phenomena is still an enormous challenge.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Aug 06, 2019
- Source ID
- W911NF1910408
Entities
People
- Jean-luc Guermond
Organizations
- Army Contracting Command
- Texas A&M University
- United States Army