A generalization of the Runge–Kutta iteration

Abstract

a b s t r a c tIterative 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 1989]) we compare them with those that we obtain when we optimize for an integrated 2-grid V -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.

Multi-stage solvers optimized for damping and propagation

“…We compare the optimal coefficients that were found in [8,5] and those obtained by optimizing I in Eq. (28).…”

“…In [8] we showed that, in order to obtain optimal performance, the smoother had to be tuned to be as complementary as possible to the defect correction.…”

“…A model for the combined transmittance of restriction, defect correction and prolongation was established in [8], which we call µ DC :…”

“…In this paper we refine the model established in [8] by also looking at the propagation characteristics. Again we are only interested in reaching convergence quickly using a multi-grid scheme, and therefore focus on the multi-stage scheme for which the 2-grid cycle will remove all frequencies as quickly as possible, either by propagation or by damping.…”

“…In [8] we chose to look at a complete 2-grid cycle which closely modeled the interaction between multi-stage solver, defect correction, restriction and prolongation. Multi-stage solvers with an improved performance were found, although their smoothing capacity was sacrificed to some extent.…”