Cole-Hopf transformation as numerical tool for the Burgers equation. (English) Zbl 1187.65116

Ohwada, Taku

Appl. Comput. Math. 8, No. 1, 107-113 (2009).

Summary: A numerical method solving the Burgers equation via the diffusion equation is proposed. The time variation of the numerical solution is given by a rational function. The coefficient of the polynomials in the denominator and numerator are determined by simple algebra. In the case of vanishingly small viscosity, the numerical method becomes shock-capturing only by increasing the viscosity to the order of mesh spacing locally around shocks. This procedure can be done automatically.

MSC:

65M70	Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35Q53	KdV equations (Korteweg-de Vries equations)

Keywords:

Burgers equation; Cole-Hopf transformation; shock capturing; numerical examples; diffusion equation