Symmetry-preserving discretizations of arbitrary order on structured curvilinear grids

Abstract

Mathematical descriptions of flow phenomena usually come in the form of partial differential equations. The differential operators used in these equations may have properties such as symmetry, skew-symmetry, positive or negative (definite)-ness. Symmetry-preserving methods are such that the discretized form of the continuous differential operator exhibits the same properties as the continuous operator itself. The use of symmetry-preserving discretizations makes it possible to construct discrete models which allow all the manipulations needed to prove stability and (discrete) conservation properties in the same way they were proven in the original continuous model. Furthermore, these methods allow a discretization of the continuous adjoint which is at the same time the discrete adjoint of the discrete forward model. Such adjoint models are not harder to code than the discrete forward model. This paper presents a new symmetry-preserving discretization of arbitrary order on curvilinear structured grids. The key idea is to use mutually-adjoint sampling and interpolation operators to switch between the continuous and discrete operator. The novelty of this work is that it combines three important requirements for discretizations: first, the symmetry-preserving discretization is made for arbitrary order of accuracy; second, the method works for every structured curvilinear mesh; and third, the method can be applied to every continuous operator. This paper is the first in a series of papers that gradually extends the theory to a general approach.