Efficient implementation of space-time adaptive ADER-DG finite element method with LST-DG predictor and a posteriori sub-cell WENO finite-volume limiting for simulation of non-stationary compressible multicomponent reactive flows

Abstract

The present work is devoted to the study of efficient implementation of spacetime adaptive ADER finite element discontinuous Galerkin method with a posteriori correction technique of solutions on subcells by the finite-volume ADER-WENO limiter scheme for simulation of non-stationary compressible multicomponent reactive flows. The multicomponent and reaction properties of the flow are considered in the form of convection-reaction equations. Therefore an effective scheme of splitting the original nonlinear system of algebraic equations of LST-DG predictor was developed to obtain an efficient iterative solution method. This approach is based on the use of the linearity property of the system of equations for the discrete spacetime solution for the system of convection equations, which makes it possible to pre-compute the inverse matrices, and allows the vectorized calculation of the transfer of individual components. The change in the concentrations of the components in the convection-reaction equations and the energy yield in the gas-dynamic equations due to the reactions are calculated using the global iterations. The iterative scheme for the original essentially nonlinear system of equations is split into a sequence of smaller systems. The approach is well applicable for the problems, especially in the case of a large number of components and reactions in the flow.

High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes

“…Among the different approaches available in literature, as those inspired to Cockburn and Shu [46,47], based on the use of a total variation bounded limiter, or the moment limiters [113], the artificial viscosity procedures [143] WENO-type limiters [149,150], or gradient-based limiters [118,116], we have selected the so-called a posteriori subcell finite volume (FV) limiter. This type of limiter is based on the MOOD approach [45,124], which has already been successfully applied in the ALE finite volume framework in [26,25] and in the discontinuous Galerkin case in [156,157,52,98,166,130,152,148] and, with a notation similar to the one used here, in [74,175,70,174,108,80,85]. We finally remark that shockcapturing techniques, based on subcell finite volume schemes, can also be applied in a predictive (a priori) fashion, for example as in [156,157,11,141,87].…”

“…This representation is then inserted in the weak formulation of the PDE and most commonly a semi-discrete approach is adopted which evolves the data in time following the method of lines, for example using a single-step multistage scheme like a high order Runge-Kutta time integrator [33]. In this work, we adopt a different technique for the time evolution, the ADER approach (Arbitrary high order DErivative Riemann problem), introduced in [133,164,162], then reworked in its modern formulation in the seminal paper [66], and widely used in literature (we cite here just a few recent works that span a wide range of technical improvements, analytical results, and applications [31,99,32,23,44,148,131,75]). ADER methods make use of a predictorcorrector technique to obtain uniform high order of accuracy in space and in time through a one-step fully discrete procedure which works on data in the form of spacetime high order polynomials.…”