High-order accurate finite-difference methods for partial differential equations on complex geometry

Abstract

We devise two high-order accurate finite-difference methods to solve scalar partial differential equations (PDEs), in second-order form, on curvilinear and overset grids. The first method addresses high-order accurate boundary closures. The second method develops high-order accurate fast algorithms for the wave equation on complex geometry. A high-order accurate boundary treatment through centered differences requires solution values on ghost cells, added beyond the bounds of the physical domain. To approximate the solution at such ghost values, traditional centered approaches utilize direct discretization of compatibility boundary conditions (CBCs)— generated from the expression of the boundary conditions and the PDE. Centered approaches rival one-sided approximations due to attractive stability and accuracy properties. Nevertheless, as the order of accuracy increases, some PDE problems exhibit involved algebra limiting the automation to high-orders of accuracy. Furthermore, discrete equation systems to evaluate the solution at the ghost points often couple tangentially impacting the efficiency of explicit time-dependent schemes, or schemes that employ iterative methods. To assuage such challenges at the boundary, we introduce the local compatibility boundary conditions (LCBC) method. The LCBC method computes solution values at ghost cells through a local interpolating polynomial. Interior values, boundary values, and compatibility boundary conditions determine the coefficients of the polynomial. The LCBC method reduces down to a formula describing solution approximations at ghost points in terms of interior and data values on a compact stencil, equal in width to that of the interior scheme. We give algorithms itemizing the LCBC procedure to an arbitrary order of accuracy. The local nature of the procedure avoids the algebraic difficulty and tangential coupling that occurs from the direct discretization of CBCs. We prove odd/even symmetry of the LCBC procedure for the unforced wave, heat and Laplace equations, with homogenous Dirichlet/Neumann BCs on Cartesian grids. Using symmetry, we demonstrate stability of the LCBC method with the high-order accurate Modified Equation scheme for the wave equation under a time-step restriction independent of the order of accuracy. A variety of numerical results on 2D Cartesian and curvilinear grids demonstrate accuracy up to order 6. On curvilinear and overset grids, algebraic complexity limits the usage of the high- order accurate Modified Equation (ME) scheme for the wave equation. While the ME scheme achieves high-order accuracy in both time and space in a single step, it involves taking repeated powers of the variable-coefficient spatial operator of the PDE. Traditional approaches expand such powers and discretize them using centered differences. As the order of accuracy increases, the operation count magnifies and impacts efficiency. We thus introduce the fast, high-order accurate factored modified equation (FAME) schemes. The FAME algorithms involve stages where second-order accurate differences of even derivatives of the solution are prepared to assemble high-order accurate differences in the final stage. This hierarchical procedure avoids the direct expansion of powers of the PDE spatial operator, and subsequent high-order accurate discretization. We demonstrate the superiority of the FAME schemes in terms of flop count, storage requirements and computational performance by comparing them to traditional ME algorithms. A variety of numerical examples employs the FAME schemes, along with the LCBC method, to demonstrate accuracy up to order 8 on 2D and 3D curvilinear and overset grids.