Generalized Summation-by-Parts Operators for the Second Derivative with Variable Coefficients
For computational scientists solving PDEs, this provides a more general framework for constructing SBP operators for second derivatives with variable coefficients, though the improvement is incremental over existing methods.
This work extends summation-by-parts (SBP) operators to second derivatives with variable coefficients, enabling stable and conservative discretizations for PDEs with mixed derivatives. The proposed operators achieve comparable accuracy to applying the first-derivative operator twice but with improved efficiency and flexibility for nonuniform grids.
The comprehensive generalization of summation-by-parts of Del Rey Fernández et al.\ (J. Comput. Phys., 266, 2014) is extended to approximations of second derivatives with variable coefficients. This enables the construction of second-derivative operators with one or more of the following characteristics: i) non-repeating interior stencil, ii) nonuniform nodal distributions, and iii) exclusion of one or both boundary nodes. Definitions are proposed that give rise to generalized SBP operators that result in consistent, conservative, and stable discretizations of PDEs with or without mixed derivatives. It is proven that such operators can be constructed using a correction to the application of the first-derivative operator twice that is the same as used for the constant-coefficient operator. Moreover, for operators with a repeating interior stencil, a decomposition is proposed that makes the application of such operators particularly simple. A number of novel operators are constructed, including operators on pseudo-spectral nodal distributions and operators that have a repeating interior stencil, but unequal nodal spacing near boundaries. The various operators are compared to the application of the first-derivative operator twice in the context of the linear convection-diffusion equation with constant and variable coefficients.