High-order accurate implicit time marching scheme for solving compressible Navier–Stokes equations based on temporal reconstruction

Abstract

This paper presents a new family of high-order implicit time marching schemes based on direct integration and temporal reconstruction (DITR). These schemes can be used to solve the system of ordinary differential equations (ODEs) arising from semi-discretization of partial differential equations (PDEs) such as the compressible Navier–Stokes equations. DITR methods can be constructed with third- and fourth-order temporal accuracy in a straightforward fashion, and require fewer stages than some popular implicit Runge–Kutta schemes. Some DITR ODE integrators can achieve $A$-stability or $L$-stability. We present a matrix-free iteration method for solving the DITR equations, which ensures that DITR can be efficiently implemented in practical applications. The linear stability is realized by choosing an appropriate preconditioning parameter. The numerical results demonstrate that DITR methods can achieve high-order of accuracy with comparatively low computational cost. When achieving the same level of errors in numerical solutions, some DITR methods use significantly smaller amounts of time compared with the popular ESDIRK4 method.