A Fourth Order Semi-Implicit Runge-Kutta Method for the Compressible Euler Equations

Abstract

Efficient time integration is of the utmost concern for unsteady flow computations. Within this context two semi-implicit Runge-Kutta methods of 3rd- and 4th-order and their application to two-dimensional compressible inviscid flows are presented. The semi-implicit methods developed here are expected to be considerably more stable than fully explicit methods, and more efficient than fully implicit methods. This is achieved by constructing an explicit 3rd-order Runge-Kutta method with the property that an arbitrary stabilizing implicit term may be added at each stage without degrading the order of convergence. This feature allows one to apply implicit operators based on approximate Jacobians, the choice of operator being made only with regard to stability, efficiency and storage requirements. A corresponding 4th-order method is obtained by linear Richardson extrapolation on the original 3rd order scheme, and both schemes are supplied with a local error estimation and time-step control. Numerical examples are performed on various cases using the DLR TAU-Code, a finite volume RANS solver, with a Lower-Upper Symmetric Gauss-Seidel implicit operator. It demonstrated that both semi-implicit schemes are significantly more stable than the corresponding explicit schemes, and in addition they are able to outperform standard fully implicit methods by up to a factor of 10 in terms of CPU time for given accuracy.