论文标题
一种求解汉堡方程的类似数值方法
An integral-like numerical approach for solving Burgers' equation
论文作者
论文摘要
采用非常规的方法来求解一维汉堡的方程。它基于样条多项式插值和HOPF-COLE转换。泰勒的扩展用于近似转换中的指数项，然后将简化方程的分析解决方案离散化以形成数值方案，涉及各种特殊功能。派生的方案是显式的，可以适应并行计算。但是，某些类型的边界条件不能直接指定。使用三个测试用例来检查其准确性，稳定性和并行的可伸缩性。在准确性的方面，采用的方案和五骨样条插插的表现同样好，设法减少了$ \ ell _ {1} $，$ \ ell_ {2} $和$ \ ell _ {\ ell _ {\ ell_ {\ ell_ {\ elfty} $错误规范，降低了$ 10^{-4} $ $ 10^{-4} $。由于转换，其稳定性条件$νδT/ΔX^2> 0.02 $包括粘度/扩散系数$ν$。从条件来看，即使在网格间距$ΔX$很小的情况下，这些方案也可以以较大的时间步长$ΔT$运行。这些特征表明，该方法比用于研究目的更适合操作用途。
An unconventional approach is applied to solve the one-dimensional Burgers' equation. It is based on spline polynomial interpolations and Hopf-Cole transformation. Taylor expansion is used to approximate the exponential term in the transformation, then the analytical solution of the simplified equation is discretized to form a numerical scheme, involving various special functions. The derived scheme is explicit and adaptable for parallel computing. However, some types of boundary condition cannot be specified straightforwardly. Three test cases were employed to examine its accuracy, stability, and parallel scalability. In the aspect of accuracy, the schemes employed cubic and quintic spline interpolation performs equally well, managing to reduce the $\ell_{1}$, $\ell_{2}$ and $\ell_{\infty}$ error norms down to the order of $10^{-4}$. Due to the transformation, their stability condition $νΔt/Δx^2 > 0.02$ includes the viscosity/diffusion coefficient $ν$. From the condition, the schemes can run at a large time step size $Δt$ even when grid spacing $Δx$ is small. These characteristics suggest that the method is more suitable for operational use than for research purposes.
