Solution of the generalized Riemann problem for advection–reaction equations
Solution of the generalized Riemann problem for advection–reaction equations
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DOI：
10.1098/rspa.2001.0926
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发表时间：
2002-02
期刊：
影响因子：
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通讯作者：
E. Toro;V. Titarev
中科院分区：
文献类型：
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作者：
E. Toro;V. Titarev
给出了求解对流-反应型偏微分方程广义Riemann问题的一种方法。黎曼问题在这里的推广有两个方面。首先，控制方程包括非线性平流项和反作用项，其次，初始条件由两个任意但无穷可微的函数组成，这一假设与双曲型守恒律的分段光滑解是一致的。该方法在足够小的时间内局部有效，将非齐次非线性方程的广义Riemann问题的解归结为求解齐次平流方程的一系列常规Riemann问题的初始条件的空间导数。通过模型对流-反应方程、非齐次Burgers方程和变床高的非线性浅水方程说明了该方法。
We present a method for solving the generalized Riemann problem for partial differential equations of the advection–reaction type. The generalization of the Riemann problem here is twofold. Firstly, the governing equations include nonlinear advection as well as reaction terms and, secondly, the initial condition consists of two arbitrary but infinitely differentiable functions, an assumption that is consistent with piecewise smooth solutions of hyperbolic conservation laws. The solution procedure, local and valid for sufficiently small times, reduces the solution of the generalized Riemann problem of the inhomogeneous nonlinear equations to that of solving a sequence of conventional Riemann problems for homogeneous advection equations for spatial derivatives of the initial conditions. We illustrate the approach via the model advection–reaction equation, the inhomogeneous Burgers equation and the nonlinear shallow–water equations with variable bed elevation.