A method for enhancing the stability and robustness of explicit schemes in astrophysical fluid dynamics

Abstract

A method for enhancing the stability and robustness of explicit schemes in computational fluid dynamics is presented. The method is based in reformulating explicit schemes in matrix form, which cane modified gradually into semi or strongly-implicit schemes. From the point of view of matrix-algebra, explicit numerical methods are special cases in which the global matrix of coefficients is reduced to the identity matrix $I$. This extreme simplification leads to severer stability range, hence of their robustness. In this paper it is shown that a condition, which is similar to the Courant-Friedrich-Levy (CFL) condition can be obtained from the stability requirement of inversion of the coefficient matrix. This condition is shown to be relax-able, and that a class of methods that range from explicit to strongly implicit methods can be constructed, whose degree of implicitness depends on the number of coefficients used in constructing the corresponding coefficient-matrices. Special attention is given to a simple and tractable semi-explicit method, which is obtained by modifying the coefficient matrix from the identity matrix $I$ into a diagonal-matrix $D$. This method is shown to be stable, robust and it can be applied to search for stationary solutions using large CFL-numbers, though it converges slower than its implicit counterpart. Moreover, the method can be applied to follow the evolution of strongly time-dependent flows, though it is not as efficient as normal explicit methods. In addition, we find that the residual smoothing method accelerates convergene toward steady state solutions considerably and improves the efficiency of the solution procedure.