Order Preserving Interpolation for Summation-by-Parts Operators at Non-Conforming Grid Interfaces
For researchers using finite difference methods for PDEs with second derivatives, this provides a way to handle non-conforming interfaces without losing accuracy, solving a known bottleneck.
This work addresses the order reduction issue at non-conforming grid interfaces for summation-by-parts finite difference methods. By introducing order preserving interpolation operators, the proposed method maintains the global convergence rate without sacrificing stability, achieving the same accuracy as conforming interfaces.
We study non-conforming grid interfaces for summation-by-parts finite difference methods applied to partial differential equations with second derivatives in space. To maintain energy stability, previous efforts have been forced to accept a reduction of the global convergence rate by one order, due to large truncation errors at the non-conforming interface. We avoid the order reduction by generalizing the interface treatment and introducing order preserving interpolation operators. We prove that, given two diagonal-norm summation-by-parts schemes, order preserving interpolation operators with the necessary properties are guaranteed to exist, regardless of the grid-point distributions along the interface. The new methods retain the stability and global accuracy properties of the underlying schemes for conforming interfaces.