High-order implicit-explicit multi-block method for hyperbolic PDEs

Computational Fluid Dynamics (CFD) consists of numerically solving the governing partial differential equations of fluid flow, known as the Navier-Stokes equations. These equations are time dependent and nonlinear, requiring robust numerical methods and significant computational power to obtain accurate solutions. This work seeks to explore and improve the current time-stepping schemes used in CFD in order to reduce overall computational time. A high-order scheme has been developed using a combination of implicit and explicit (IMEX) time-stepping Runge-Kutta (RK) schemes with the intent of increasing the time step while maintaining numerical stability, resulting in decreased computational time. 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 both one-dimensional (1D) and two-dimensional (2D) problems. The 1D problem studied in this work is the nonlinear viscous Burger\u27s equation, a model 1D nonlinear partial differential equation. The 2D advection equation is used to study the effects of the added dimension in space of 2D problems. The method uses two different summation by parts (SBP) derivative approximations, second-order (SBP 1-2-1) and fourth-order (SBP 2-4-2) accurate schemes in space. The Dirichlet boundary conditions are imposed using the Simultaneous Approximation Term (SAT) penalty method. The 6-stage additive Runge-Kutta IMEX time integration schemes are fourth-order accurate in time. 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\u27s 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 in this work shows potential for breaking the computational domain into manageable sizes such that implicit solutions for full three-dimensional CFD simulations can be computed using direct solving methods