Dual Time-Stepping Using Second Derivatives

We present a modified formulation of the dual time-stepping technique which makes use of two derivatives in pseudo-time. This new technique retains and improves the convergence properties to the stationary solution. When compared with the conventional dual time-stepping, the method with two derivatives reduces the stiffness of the problem and requires fewer iterations for full convergence to steady-state. In the current formulation, these positive effects require that an approximation of the square root of the spatial operator is available and inexpensive.

A 3D Approximate Riemann Solver for the Euler Equations Using Flux Splitting Schemes With a Robust Multigrid

An explicit 3D approximate Riemann solver for the Euler equations is proposed using the famous shock capturing schemes with a simple cell vertex based multigrid method. A multistage Runge-Kutta time marching scheme with a local time stepping is used to achieve fast convergence to steady state. A Roe’s flux difference splitting, AUSM+, Van Leer and Steger-Warming’s flux vector splitting are implemented as base Riemann solvers with a third order flux reconstruction. It is shown that the proposed Riemann solvers accurately capture the shocks as well as reduce CPU time significantly with new multigrid.