Higher order numerical discretizations for exterior and biharmonic type PDEs

Reuse of standard preconditioners for higher-order time discretizations of parabolic PDEs

In this work we study a preconditioned iterative method for some higher–order time discretizations of linear parabolic partial differential equations. We use the Padé approximations of the exponential function to discretize in time and show that standard solution algorithms for lower–order time discretization schemes, such as Crank–Nicolson and implicit Euler, can be reused as preconditioners f...

Decoupled, Unconditionally Stable, Higher Order Discretizations for MHD Flow Simulation

We propose, analyze, and test a new MHD discretization which decouples the system into two Oseen problems at each timestep yet maintains unconditional stability with respect to the time step size, is optimally accurate in space, and behaves like second order in time in practice. The proposed method chooses a parameter θ ∈ [0, 1], dependent on the viscosity ν and magnetic diffusivity νm, so that...

Domain Decomposition Preconditioners for Higher-Order Discontinuous Galerkin Discretizations

Aerodynamic flows involve features with a wide range of spatial and temporal scales which need to be resolved in order to accurately predict desired engineering quantities. While computational fluid dynamics (CFD) has advanced considerably in the past 30 years, the desire to perform more complex, higher-fidelity simulations remains. Present day CFD simulations are limited by the lack of an effi...

Multigrid Solution of Automatically Generated High-Order Discretizations for the Biharmonic Equation

In this work, we use a symbolic algebra package to derive a family of nite diierence approximations for the biharmonic equation on a 9 point compact stencil. The solution and its rst derivatives are carried as unknowns at the grid points. Dirichlet boundary conditions are thus incorporatednaturally. Since the approximations use the 9 point compact stencil, no special formulas are needed near th...

AFRL-OSR-VA-TR-2015-0114 Tailoring high order time discretizations for use with spatal discretizations of hyperbolic PDEs