High-Order Semi-implicit Schemes for Evolutionary Partial Differential Equations with Higher Order Derivatives

Abstract The aim of this work is to apply a semi-implicit (SI) strategy in an implicit-explicit (IMEX) Runge–Kutta (RK) setting introduced Boscarino et al. (J Sci Comput 68:975–1001, 2016) sequence 1D time-dependent partial differential equations (PDEs) with high order spatial derivatives. This gives great flexibility treat these equations, and allows the construction simple linearly implicit schemes without any Newton’s iteration. Furthermore, SI IMEX-RK so designed does not need severe time step restriction that usually one has using explicit methods for stability, i.e. $$\Delta t = {\mathcal {O}}(\Delta t^k)$$ Δ t = O ( k ) k th ( $$k \ge 2$$ ≥ 2 ) PDEs. For space discretization, combined finite difference schemes. We illustrate effectiveness many applications dissipative, dispersive biharmonic-type equations. Numerical experiments show proposed are stable can achieve optimal orders accuracy.