非守恒形式非线性反应扩散问题的高阶松弛格式
High order relaxed schemes for nonlinear reaction diffusion problems in nonconservative form
AI总结：
本文提出一种非标准松弛格式用于非守恒形式扩散方程，结合ENO/WENO空间离散和IMEX时间积分实现高阶精度，并通过多个一维和二维算例验证了格式的有效性。
摘要：
最近提出了针对偏微分方程的不同松弛近似，包括守恒律、Hamilton-Jacobi方程、对流扩散问题、气体动力学问题。本文聚焦于用于非线性反应扩散方程数值近似的扩散松弛格式。我们选择了一种非标准松弛格式，允许处理非守恒形式的扩散方程。还包括了与守恒方程情况下传统方法的比较。通过将ENO和WENO格式用于空间离散，IMEX格式用于时间积分，获得了高阶方法，其中隐式部分可以以线性代价显式求解。为了说明所提数值格式的高精度和良好性质，包括退化情况，我们考虑了一维和二维的各种例子：Fisher-Kolmogoroff方程、多孔Fisher方程和具有强吸收的多孔介质方程。此外，我们展示了一个描述动物种群扩散和定居的生态模型PDE系统的测试。
英文摘要：
Different relaxation approximations to partial differential equations, including conservation laws, Hamilton-Jacobi equations, convection-diffusion problems, gas dynamics problems, have been recently proposed. The present paper focuses onto diffusive relaxed schemes for the numerical approximation of nonlinear reaction diffusion equations. We choose here a nonstandard relaxation scheme that allow the treatment of diffusion equations in their nonconservative form. A comparison with the traditional approach in the case of conservative equations is also included. High order methods are obtained by coupling ENO and WENO schemes for space discretization with IMEX schemes for time integration, where the implicit part can be explicitly solved at a linear cost. To illustrate the high accuracy and good properties of the proposed numerical schemes, also in the degenerate case, we consider various examples in one and two dimensions: the Fisher-Kolmogoroff equation, the porous-Fisher equation and the porous medium equation with strong absorption. Moreover we show a test on a system of PDEs that describe an ecological model for the dispersal and settling of animal populations.
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