瞬态扩散方程最大值原理与非负约束的数值方法
A numerical methodology for enforcing maximum principles and the non-negative constraint for transient diffusion equations
AI总结：
针对瞬态各向异性扩散方程，提出一种基于Rothe方法（水平线法）的数值格式，在一般计算网格上无时间步限制地满足最大值原理和非负约束，并通过数值算例验证了精度和收敛性。
摘要：
瞬态扩散方程出现在工程和应用科学的许多分支（例如传热和传质）中，是抛物型偏微分方程。众所周知，在输入数据的某些假设下，这些方程满足重要的数学性质，如最大值原理和非负约束，这些性质在数学建模中具有重要意义。然而，现有的数值格式通常不满足最大值原理和非负约束。本文提出了一种强制瞬态各向异性扩散方程满足最大值原理和非负约束的方法。采用水平线法（也称为Rothe方法），首先对时间进行离散。这导致在每个离散时间层上求解带有衰减项的稳态各向异性扩散方程。所提出的瞬态各向异性扩散方程方法将在一般计算网格上满足最大值原理和非负约束，且对时间步长没有额外限制。我们通过代表性数值算例说明了所提格式的性能和精度。我们还对所提方法进行了数值收敛性分析。为了比较，我们还给出了标准单场半离散格式和流行软件包的结果，这些结果都会违反最大值原理和非负约束。
英文摘要：
Transient diffusion equations arise in many branches of engineering and applied sciences (e.g., heat transfer and mass transfer), and are parabolic partial differential equations. It is well-known that, under certain assumptions on the input data, these equations satisfy important mathematical properties like maximum principles and the non-negative constraint, which have implications in mathematical modeling. However, existing numerical formulations for these types of equations do not, in general, satisfy maximum principles and the non-negative constraint. In this paper, we present a methodology for enforcing maximum principles and the non-negative constraint for transient anisotropic diffusion equation. The method of horizontal lines (also known as the Rothe method) is applied in which the time is discretized first. This results in solving steady anisotropic diffusion equation with decay equation at every discrete time level. The proposed methodology for transient anisotropic diffusion equation will satisfy maximum principles and the non-negative constraint on general computational grids, and with no additional restrictions on the time step. We illustrate the performance and accuracy of the proposed formulation using representative numerical examples. We also perform numerical convergence of the proposed methodology. For comparison, we also present the results from the standard single-field semi-discrete formulation and the results from a popular software package, which all will violate maximum principles and the non-negative constraint.
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