High-order Locally A-stable Implicit Schemes for Linear ODEs

Accurate simulations of wave propagation in complex media like Earth subsurface can be performed with a reasonable computational burden by using hybrid meshes stuffing fine and coarse cells. Locally implicit time discretizations are then of great interest. They indeed allow using unconditionally stable Schemes in the regions of computational domain covered by small cells. The receivable values of the time step are then increased which reduces the computational costs while limiting the dispersion effects. In this work we construct a method that combines optimized explicit Schemes and implicit Schemes to form locally implicit schemes for linear ODEs, including in particular semi-discretized wave problems that are considered herein for numerical experiments. Both the explicit and implicit schemes used are one-step methods constructed using their stability function. The stability function of the explicit schemes are computed by maximizing the time step that can be chosen. The implicit schemes used are unconditionally stable and do not necessary require the same number of stages as the explicit schemes. The performance assessment we provide shows a very good level of accuracy for locally implicit schemes. It also shows that a locally implicit scheme is a good compromise between purely explicit and purely implicit schemes in terms of computational time and memory usage.