An Integral-like Numerical Approach for Solving Burgers’ Equation

Abstract

The Burgers’ equation, commonly appeared in the study of turbulence, fluid dynamics, shock waves, and continuum mechanics, is a crucial part of the dynamical core of any numerical weather model, influencing simulated meteorological phenomena. While past studies have suggested several robust numerical approaches for solving the equation, many are too complicated for practical adaptation and too computationally expensive for operational deployment. This paper introduces an unconventional approach based on spline polynomial interpolations and the Hopf-Cole transformation. Using Taylor expansion to approximate the exponential term in the Hopf-Cole transformation, the analytical solution of the simplified equation is discretized to form our proposed scheme. The scheme is explicit and adaptable for parallel computing, although certain types of boundary conditions need to be employed implicitly. Three distinct test cases were utilized to evaluate its accuracy, parallel scalability, and numerical stability. In the aspect of accuracy, the schemes employed cubic and quintic spline interpolation perform equally well, managing to reduce the <i>ӏ</i><sub>1</sub>, <i>ӏ</i><sub>2</sub>, and <i>ӏ</i><sub>∞</sub> error norms down to the order of 10<sup>−4</sup>. Parallel scalability observed in the weak-scaling experiment depends on time step size but is generally as good as any explicit scheme. The stability condition is <i>ν</i>∆<i>t</i>/∆<i>x</i><sup>2</sup> > 0.02, including the viscosity coefficient <i>ν</i> due to the Hopf-Cole transformation step. From the stability condition, the schemes can run at a large time step size ∆<i>t</i> even when using a small grid spacing ∆<i>x</i>, emphasizing its suitability for practical applications such as numerical weather prediction.