Research Summaries

Back Efficient High-Order Time-Integrators for High-Order Discretization Methods

Fiscal Year 2014
Division Graduate School of Engineering & Applied Science
Department Applied Mathematics
Investigator(s) Giraldo, Francis X.
Sponsor Air Force Office of Scientific Research (Air Force)
Summary We propose to develop high-order accurate and highly efficient time-integration methods for local high-order spatial discretizations of hyperbolic partial differential equations typically found in fluid flow problems; we plan to construct a hierarchy of time-integrators such as: explicit, implicit-explicit, extrapolation, multi-rate, and fully-implicit methods. We will focus our efforts on the numerical solution of the shallow water, Euler, and compressible Navier-Stokes equations. For this class of differential equations, local high-order methods such as finite volumes, finite elements, and finite differences have shown to give excellent solution accuracy while scaling well on the current class of multi-core parallel computers and should also perform well on graphical processing units (GPUs). We propose to use spectral element and discontinuous Galerkin methods as representative of the state-of-the-art local high-order methods. Using these methods to discretize the partial differential equations we will first solve the resulting system of ordinary differential equations via high-order explicit time-integration methods that are both stable and efficient when using large time-steps. These optimal explicit time-integrators will then be employed to construct more efficient time-integrators using the Implicit-Explicit (IMEX) approach (this method is also known as the semi-implicit method). The challenge will be to devise high-order IMEX methods that are high-order accurate, at least A-stable (stable on the entire left-hand plane), and efficient. We also propose to implement extrapolation methods (e.g., Gragg extrapolation) not only in an explicit approach but also in a multi-rate approach. Finally, we propose to study fully-implicit methods. It should be mentioned that while IMEX and fully-implicit methods (FIM) have been used for high-order finite elements and spectral element methods, there has been little to no development for Godunov-type methods such as discontinuous Galerkin methods. In summary, we propose to develop the following hierarchy of high-order time-integrators:
1. explicit, strong-stability preserving (SSP) and explicit non-SSP,
2. IMEX (these methods have the advantage of possessing a Schur complement that can be extracted and exploited),
3. extrapolation methods (both in explicit and implicit forms), and
4. fully-implicit method (with approximate Jacobians, i.e., Rosenbrock, and Jacobian-free methods).
It should be mentioned that the success of both the IMEX and FIM will be highly dependent on constructing good preconditioners and iterative solvers. Thus a significant amount of effort must be devoted to the construction of preconditioners specifically designed for our spatial operators. We will measure the efficiency of the methods and models both on serial and parallel computers.
Publications Publications, theses (not shown) and data repositories will be added to the portal record when information is available in FAIRS and brought back to the portal
Data Publications, theses (not shown) and data repositories will be added to the portal record when information is available in FAIRS and brought back to the portal