Investigating the use of a rational Runge Kutta method for transport modelling

An unconditionally stable explicit time integrator has recently been developed for parabolic systems of equations. This rational Runge Kutta (RRK) method, proposed by Wambecq¹ and Hairer², has been applied by Liu et al.³ to linear heat conduction problems in a time-partitioned solution context. An important practical question is whether the method has application for the solution of (nearly) hyperbolic equations as well.In this paper the RRK method is applied to a nonlinear heat conduction problem, the advection-diffusion equation, and the hyperbolic Buckley-Leverett problem. The method is, indeed, found to be unconditionally stable for the linear heat conduction problem and performs satisfactorily for the nonlinear heat flow case. A heuristic limitation on the utility of RRK for the advection-diffusion equation arises in the Courant number; for the second-order accurate one-step two-stage RRK method, a limiting Courant number of 2 applies. First order upwinding is not as effective when used with RRK as with Euler one-step methods. The method is found to perform poorly for the Buckley-Leverett problem.