Extended Skew-Symmetric Form for Summation-by-Parts Operators and Varying Jacobians
This work provides a theoretical foundation for stable high-order methods on curvilinear grids, benefiting computational fluid dynamics and conservation law simulations.
The paper extends summation-by-parts (SBP) operators to enable entropy stable split-form schemes for Burgers' equation and achieves stability on curvilinear grids with dense norms, overcoming prior limitations of finite difference methods.
A generalised analytical notion of summation-by-parts (SBP) methods is proposed, extending the concept of SBP operators in the correction procedure via reconstruction (CPR), a framework of high-order methods for conservation laws. For the first time, SBP operators with dense norms and not including boundary points are used to get an entropy stable split-form of Burgers' equation. Moreover, overcoming limitations of the finite difference framework, stability for curvilinear grids and dense norms is obtained for SBP CPR methods by using a suitable way to compute the Jacobian.