$$H$$ -adaptive Mesh Method with Double Tolerance Adaptive Strategy for Hyperbolic Conservation Laws

Abstract

The numerical method used to solve hyperbolic conservation laws is often an explicit scheme. As a commonly used technique to improve the quality of numerical simulation, the \(h\)-adaptive mesh method is adopted to resolve sharp structures in the solution. Since the computational costs of altering the mesh and solving the PDEs are comparable, too often the mesh adaption triggered may bring down the overall efficiency of solving hyperbolic conservation laws using \(h\)-adaptive mesh method. In this paper, we propose a so-called double tolerance adaptive strategy to optimize the overall numerical efficiency by reducing the number of mesh adaptions, as well as preserving the quality of the numerical solution. Numerical results are presented to demonstrate the robustness and effectiveness of our \(h\)-adaptive algorithm using the double tolerance adaptive strategy.