On coordinate transformations for summation-by-parts operators
Many problems in computational fluid dynamics require very accurate spatial resolution to correctly model high frequency phenomena. The author examines properties of a special class of high order finite difference operators obeying a summation-by-parts rule (SBP-operators) developed by \textit{H. O. Kreiss} and \textit{G. Scherer} in [Finite element and finite difference methods for hyperbolic partial differential equations. Math. Aspects Finite Elem. Partial Differ. Equ., Proc. Symp. Madison 1974, 195--212 (1974; Zbl 0355.65085) and On the existence of energy estimates for difference approximations for hyperbolic systems (1977)]. These norms are of a non-diagonal form. The main advantage of SBP-operators is that they simplify the procedure to obtain energy estimates. To accurately model the influence of realistic geometries, it is essential to be able to use the SBP-operators on curvilinear grids. In the paper accurate coordinate transformation operators are analyzed using the linear advection equation as a model problem. So, the objective of the paper is to study how coordinate transformations influence high order SBP schemes and if possible, construct an accurate SBP preserving coordinate transformation operator. But, as is stated in Theorem 3.2, the conditions for a sufficiently accurate SBP preserving coordinate operators are not possible to fulfill. Thus, either the stability or the accuracy conditions must be relaxed. If the accuracy conditions are kept (using a block norm) there is no systematic way of proving stability and the location of the eigenvalues has to be checked for each problem. Further, there is nothing that guarantees stability when the number of grid points are increased. If the accuracy conditions are relaxed, a diagonal norm scheme was shown to be the only choice that fulfill the remaining stability conditions making it possible to treat the boundaries in a stable and systematic manner. The contents of the paper report are divided as follows: in Section 2 the linear advection equation is considered and an energy estimate is derived; the equation is discretized on a non-equidistant grid. A general coordinate transformation operator is analyzed in Section 3 which also contains numerical experiments. In Section 4, conclusions are drawn.
scientific article; zbMATH DE number 2069185