Generalized Summation-by-Parts Operators for the Second Derivative with Variable Coefficients

TL;DR

This paper extends summation-by-parts operators to second derivatives with variable coefficients, enabling flexible, stable, and conservative discretizations for PDEs with diverse boundary and nodal configurations.

Contribution

It introduces a generalized framework for SBP operators for second derivatives with variable coefficients, including novel constructions and simplifications for practical implementation.

Findings

Constructed new SBP operators with nonuniform nodes and boundary exclusions.

Proven correction method for constant and variable coefficient operators.

Compared operators in convection-diffusion context demonstrating effectiveness.

Abstract

The comprehensive generalization of summation-by-parts of Del Rey Fern\'andez 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.