A generalization of the Runge–Kutta iteration

Iterative solvers in combination with multi-grid have been used extensively to solve large algebraic systems. One of the best known is the Runge–Kutta iteration. We show that a generally used formulation [A. Jameson, Numerical solution of the Euler equations for compressible inviscid fluids, in: F. Angrand, A. Dervieux, J.A. Désidéri, R. Glowinski (Eds.), Numerical Methods for the Euler Equations of Fluid Dynamics, SIAM, Philadelphia, 1985, pp. 199–245] does not allow to form all possible polynomial transmittance functions and we propose a new formulation to remedy this, without using an excessive number of coefficients.After having converted the optimal parameters found in previous studies (e.g. [B. Van Leer, C.H. Tai, K.G. Powell, Design of optimally smoothing multi-stage schemes for the Euler equations, AIAA Paper 89–1923, 1989]) we compare them with those that we obtain when we optimize for an integrated 2-grid VV-cycle and show that this results in superior performance using a low number of stages. We also propose a variant of our new formulation that roughly follows the idea of the Martinelli–Jameson scheme [A. Jameson, Analysis and design of numerical schemes for gas dynamics 1, artificial diffusion, upwind biasing, limiter and their effect on multigrid convergence, Int. J. Comput. Fluid Dyn. 4 (1995) 171–218; J.V. Lassaline, Optimal multistage relaxation coefficients for multigrid flow solvers. http://www.ryerson.ca/~jvl/papers/cfd2005.pdf] used on the advection–diffusion equation which that can be extended to other types. Gains in the order of 30%–50% have been shown with respect to classical iterative schemes on the advection equation. Better results were also obtained on the advection–diffusion equation than with the Martinelli–Jameson coefficients, but with less than half the number of matrix-vector multiplications.