| Author | Jason Hicken, David C. Del Rey Fernández, and David W. Zingg |
|---|---|
| Title | Multidimensional Summation-By-Parts Operators: General Theory and Application to Simplex Elements |
| Year | 2015 |
| Journal | submitted to the Journal of Computational Physics |
| Abstract | Summation-by-parts (SBP) operators offer the efficiency of finite-difference methods with the provable time stability of Galerkin finite-element methods, but they have traditionally been limited to tensor-product domains. This paper presents a definition for multidimensional SBP finite-difference operators that is a natural extension of the classical one-dimensional SBP definition. Theoretical implications of the definition are investigated for the special case of a diagonal-norm (mass) matrix, and it is shown that the operators retain the desirable properties of tensor-product SBP operators. A cubature rule with positive weights is proven to be a necessary and sufficient condition for the existence of diagonal-norm SBP operators on a particular domain. Concrete examples of multidimensional SBP operators are constructed for the triangle and tetrahedron; similarities and differences with spectral-element and spectral-difference methods are discussed. An assembly process is described that builds diagonal-norm SBP operators on a global domain from element-level operators. Numerical results of linear advection on a doubly periodic domain demonstrate the accuracy and time stability of the simplex operators. |
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