On unconditional conservation of kinetic energy by finite-difference discretizations of the linear and non-linear convection equation

Abstract

In the past the development of kinetic energy conserving finite-difference methods mostly focused on second-order accurate central methods defined on uniform grids. Nowadays the need for high-order accurate discretizations, to perform for instance accurate numerical simulations of turbulent flow, calls for the development of novel kinetic energy conserving discretization schemes. Instead of choosing a fixed basis discretization up front, in this paper a different, more general, approach is applied. For a Cartesian mesh, sets of conditions are presented such that all discretizations of the linear or non-linear convection equation which obey these conditions, unconditionally conserve kinetic energy. For the linear convection equation it is shown that on a uniform grid it is necessary and sufficient for a discretization to be central in order to be fully conservative, that is: such discretizations not only unconditionally conserve kinetic energy but also unconditionally conserve momentum. On non-uniform grids an algorithm is introduced that can be used to generate fully conservative discretizations that are at least first-order accurate. The derivation of the discretization conditions for the non-linear convection equation is performed in the two-dimensional (2D) linear case. Some examples on uniform grids and on non-uniform grids are presented. It is shown that on uniform grids no upper limit exists with respect to the accuracy of the kinetic energy conserving method. For the higher-dimensional linear and non-linear convection equation the same set of conditions, which ensure the unconditional conservation of kinetic energy, are found as in the 2D linear case. Other results too are found to be straightforward generalizations of the corresponding 2D linear results. It is shown that the fourth-order unconditionally kinetic energy conserving discretization on a staggered mesh introduced in this paper is well suited to simulate the initial development of an inviscid shear layer instability in a divergence-free flow.