Investigation and application of point implicit Runge–Kutta methods to inviscid flow problems

Abstract

We derive and investigate point implicit Runge–Kutta methods to significantly improve the convergence rate to approximate steady-state solutions of inviscid flows. It turns out that the point implicit Runge–Kutta can be interpreted as a preconditioned explicit Runge–Kutta method, where the preconditioner arises naturally as local derivative of the residual function. Moreover, many preconditioners suggested in the literature so far are identified as special case of our general ansatz. Conditions will be formulated such that explicit Runge–Kutta methods with local time stepping are equivalent to point implicit methods. In numerical examples, we will demonstrate the improved convergence rates.