Local high-order conservative space-time adaptive mesh refinement with the Runge-Kutta time integration for solving the systems of conservation laws

Abstract

Numerical simulation is widely used as a main way of researching the processes described by the systems of partial differential equations in the divergence form, like fluid dynamics, traffic flow, wave propagation. To obtain desired accuracy and resolution, the high-order numerical methods are used. In this paper we consider the Runge-Kutta (RK) time discretization, when the space discretization can be high-order finite volume method, discontinuous Galerkin method or others. Presence of discrete analogue of conservation is considered to be very important for such methods as it corresponds to the essential feature of simulated process, becoming especially vital at the discontinuous solutions. The adaptive mesh refinement (AMR) techniques are used to get the numerical solution with more resolution where needed. For the explicit numerical scheme obeying the Courant-Friedrichs-Lewy (CFL) condition, the bigger time integration step can be chosen for the coarser cells. The problem is how to reformulate the numerical scheme near the boundary between cells of different size, keeping the conservation and high-order accuracy. In this paper a new algorithm is suggested. The changes in the numerical method are made only in the vicinity of the boundary. The local character of the developed algorithm allows us to efficiently implement it at modern computer systems. Results of validation tests of developed solver are shown.