Energy properties preserving schemes for Burgers' equation

• Published in: Numerical methods for partial differential equations 2008-01, Vol.24 (1), p.41-59
• Main Authors: Anguelov, R., Djoko, J.K., Lubuma, J.M.-S.
• Format: Article
• Language: English
• Subjects: Burgers equation; kinetic energy; non-standard finite difference; nonlocal approximation; standard finite difference
• ISSN: 0749-159X; 1098-2426
• DOI: 10.1002/num.20227

The Burgers' equation, a simplification of the Navier–Stokes equations, is one of the fundamental model equations in gas dynamics, hydrodynamics, and acoustics that illustrates the coupling between convection/advection and diffusion. The kinetic energy enjoys boundedness and monotone decreasing properties that are useful in the study of the asymptotic behavior of the solution. We construct a family of non‐standard finite difference schemes, which replicate the energy equality and the properties of the kinetic energy. Our approach is based on Mickens' rule [Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994.] of nonlocal approximation of nonlinear terms. More precisely, we propose a systematic nonlocal way of generating approximations that ensure that the trilinear form is identically zero for repeated arguments. We provide numerical experiments that support the theory and demonstrate the power of the non‐standard schemes over the classical ones. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007