一维双曲守恒律的龙格-库塔控制体积间断有限元方法

RKCVDFEM for One-dimensional Hyperbolic Conservation Laws

给出数值求解一维双曲守恒律方程的新方法——龙格-库塔控制体积间断有限元方法(RKCVDFEM),其中空间离散基于控制体积有限元方法,时间离散基于二阶TVB Runge-Kutta技术,有限元空间选取为分段线性函数空间.理论分析表明,格式具有总变差有界(TVB)的性质,而且空间和时间离散形式上具有二阶精度.数值算例表明,数值解收敛到熵解并且对光滑解的收敛阶是最优的,优于龙格-库塔间断Galerkin方法(RKDGM)的计算结果.

Runge-Kutta control volume discontinuous finite element method （RKCVDFEM） is proposed to solve numerically hyperbolic conservation laws, in which space discretization is based on control volume finite element method（CVFEM） while time discretization is based on a second order accurate TVB Runge-Kutta technique. Pieeewise linear function space is chosen as finite element space. The scheme is total variation bounded（TVB） and is formally second order accurate in space and time. Numerical examples show that numerical solution converges to the entropy solution, and order of convergence is optimal for smooth solution. Compared with numerical results of Runge-Kutta discontinuous Galerkin method（RKDGM） those of RKCVDFEM are better.