Holistic discretisation ensures fidelity to Burger's equation

Abstract

I analyze a generalized Burgers' equation to introduce a new method of spatial discretization. The analysis is based upon center manifold theory so we are assured that the discretization accurately models the dynamics and may be constructed systematically. The trick to the application of center manifold theory is to divide the physical domain into small elements by introducing insulating internal boundaries which are later removed. Burgers' equation is used as an example to show how the concepts work in practice. The resulting finite difference models are shown to be significantly more accurate than conventional discretizations, particularly for highly nonlinear dynamics. This center manifold approach treats the dynamical equations as a whole, not just as the sum of separate terms—it is holistic. The techniques developed here may be used to accurately model the nonlinear evolution of quite general spatio-temporal dynamical systems.