Numerical optimization of fourth-order diagonally-implicit multistep Runge-Kutta methods

This article presents constrained numerical optimization of fourth-order L-stable multistep Runge-Kutta (MRK) methods. The methods are optimized relative to composite objective functions accounting for accuracy, internal stability, conditioning, and computational cost. Global stability properties and bounds on the coeficients are enforced through linear and nonlinear constraints. The relative bene ts of increasing the number of stages versus the number of steps is discussed, along with comparisons to implicit linear multistep (LM) and implicit Runge-Kutta (RK) methods. With the chosen objective function, the optimized MRK methods are not expected to be the most efficient. However, they do obtain a combination of properties that neither the LM or RK methods can. Furthermore, when applied to laminar ow over a circular cylinder, the optimized L-stable fourth-order two-step four-stage sti y-accurate singly-diagonally-implicit multistep Runge-Kutta method SDIMRK[4,2](4,2)L_SA_0 was the most e cient. This is partly due to the accuracy being more related to the local truncation error for this case, rather than the L2-principal error norm for which the methods were optimized. Simulation of van der Pol's equation also con rms the order properties of all methods for nonstiff and stiff problems.