Time-marching schemes for spatially high order accurate discretizations of the Euler and Navier–Stokes equations

Computational fluid dynamics (CFD) methods used for the numerical solution of the Euler and Navier–Stokes equations have been sufficiently matured and enable to perform high fidelity simulations in fluid dynamics research and engineering applications. In this review, some low-order (second order or lower) accurate space–time-domain discretization schemes that are still widely in use are reviewed first, in order to show the benefits of high order numerical schemes and the techniques for stability and error analysis. Then, popular high order spatial discretization schemes are discussed to highlight the benefits and also the challenges they impose on high order implicit time advancement. After these, we focus on the major aspects of implicit time advancement combining the Runge–Kutta methods and high order spatial discretizations that have been proven efficient to resolve unsteady flows. In addition to the construction of high order implicit Runge–Kutta schemes, more recent development concerning enhanced nonlinear stability and low-dispersion low-dissipation errors is discussed in detail for multi-physical flow phenomena. Efficient solution techniques for implicit parallel solutions on advanced high-performance computers are reviewed, such as the traditional LU-SGS and ADI methods based on the approximate factorization, the Newton iterative method with subsidiary iterations, etc. As another challenging issue, enforcement of implicit boundary conditions is also elaborated, and we focus especially on the recent developments and the benefits they offer regarding computational efficiency and accuracy.