双曲型守恒律方程的自适应间断 Galerkin 方法∗

摘要

工程实际中的许多间断问题，例如空气动力学中的激波问题，其数学模型大都是非线性双曲守恒律方程。本文在 Runge-Kutta 间断 Galerkin (RKDG)框架下，结合 h 型自适应方法处理了一维非线性守恒律方程初值问题和初边值问题。此方法不仅能准确描述间断的出现和位置，而且还能在间断附近适当加密网格，提高计算效率。最后，数值算例验证了算法的有效性。%Some discontinuous problems like the shock wave problem of aerodynamics can be described by a nonlinear hyperbolic conservation law. In this paper, we present a adap-tive discontinuous Galerkin method for the initial value and initial-boundary value problem of one-dimensional nonlinear hyperbolic conservation law. The method introduces a h-adaptive strategy in the framework of Runge-Kutta discontinuous Galerkin finite element (RKDG). Then the appearance and position of discontinuity is captured by the method, and the mesh is prop-erly refined near the discontinuity to improve calculation efficiency. Finally, the correctness of the propose results is verified by numerical examples.