High-Order Implicit-Explicit Multi-Block Time-stepping Method for Hyperbolic PDEs

摘要

This work seeks to explore and improve the current time-stepping schemes used to numerically solve model PDEs relevant to computational fluid dynamics (CFD). A high-order scheme has been developed using a combination of implicit and explicit (IMEX) time-stepping Runge-Kutta (RK) schemes, which increases the stability with respect to the time step limit. The IMEX scheme alone does not yield the desired increase in numerical stability, but when used in conjunction with an overlapping partitioned (multi-block) domain significant increase in stability is observed. To show this, the Overlapping-Partition IMEX (OP IMEX) scheme is applied to the one-dimensional viscous Burger's equation and the two-dimensional advection equation. The method uses second and fourth order accurate summation-by-parts (SBP) finite differences, and a fourth order accurate 6-stage additive Runge-Kutta IMEX time integration scheme. The Dirichlet boundary conditions are imposed using the Simultaneous Approximation Term (SAT) penalty method. An increase in numerical stability, ～65 times greater than the fully explicit scheme, is demonstrated to be achievable with the OP IMEX method applied to 1D Burger's equation. Results from the 2D, purely convective, advection equation show stability increases on the order of ～10 times the explicit scheme using the OP IMEX method. Also, the domain partitioning method developed in this work shows potential for semi-implicitly solving full three-dimensional CFD simulations using direct methods, rather than the widely used iterative methods. This domain partitioning approach achieves this by splitting the computational domain into manageable sizes, or multiple blocks, which are explicitly coupled together.