Family of symmetric bicompact schemes with spectral resolution property for hyperbolic equations

Abstract: For the numerical solution of nonstationary quasilinear hyperbolic equations, a family of symmetric semidiscrete bicompact schemes based on collocation polynomials is constructed in the one- and multidimensional cases. A dispersion analysis of semidiscrete bicompact schemes of fourth to eighth orders of accuracy in space is performed. Numerical examples are presented that demonstrate the ability of the bicompact schemes to adequately simulate wave propagation, including short waves, on highly nonuniform grids at long times. The properties of solutions of the bicompact schemes in the problem of transfer of a stepwise initial profile are also considered.Note: Research direction:Mathematical problems and theory of numerical methodsRussia

Two variants of parallel implementation of high-order accurate bicompact schemes for multi-dimensional inhomogeneous transport equation

Abstract: In this paper, we compare the efficiency of two parallel algorithms for solution of the equations of multidimensional high-order accurate bicompact schemes for a multidimensional inhomogeneous transport equation. The first algorithm is a space-marching one for computing non-factorized schemes, and the second algorithm is based on the approximate factorization of multidimensional schemes. The latter algorithm uses iterations to preserve the high (higher than second) order of accuracy of bicompact schemes in time. The convergence of these iterations is proved for a nonstationary two-dimensional and three-dimensional linear inhomogeneous transport equation with constant positive coefficients. Model computations show that the factorization scheme is preferable from the point of view of parallel implementation.Note: Research direction:Mathematical problems and theory of numerical methodsRussia