An integral-like numerical approach for solving Burgers' equation

TL;DR

This paper introduces a novel numerical method combining spline interpolation and Hopf-Cole transformation to solve Burgers' equation efficiently, with high accuracy, stability, and parallel scalability, suitable for operational applications.

Contribution

It presents an explicit, parallelizable numerical scheme based on spline interpolation and Taylor expansion of the Hopf-Cole transformation, offering improved stability and accuracy for Burgers' equation.

Findings

Achieved error norms around 10^{-4} with cubic and quintic splines.

Stable for large time steps when viscosity/diffusion coefficient is considered.

Method is suitable for operational use due to stability and scalability.

Abstract

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 $\nu \Delta t/\Delta x^2 > 0.02$ includes the viscosity/diffusion coefficient $\nu$. From the condition, the schemes can run at a large time step size $\Delta t$ even when grid spacing $\Delta x$ is small. These characteristics suggest that the method is more suitable for operational use than for research purposes.