Boundary and interface methods for energy stable finite difference discretizations of the dynamic beam equation

TL;DR

This paper compares boundary and interface methods for energy stable finite difference discretizations of the dynamic beam equation, introducing novel approaches and analyzing their accuracy and computational efficiency.

Contribution

It introduces and compares SBP-SAT, SBP-P, and hybrid SBP-SAT-P methods for the dynamic beam equation with discontinuous coefficients, including new discretizations for interface conditions.

Findings

All methods have similar accuracy.

SBP-P can be more computationally efficient.

Methods are stable and suitable for problems with discontinuous coefficients.

Abstract

We consider energy stable summation by parts finite difference methods (SBP-FD) for the homogeneous and piecewise homogeneous dynamic beam equation (DBE). Previously the constant coefficient problem has been solved with SBP-FD together with penalty terms (SBP-SAT) to impose boundary conditions. In this work we revisit this problem and compare SBP-SAT to the projection method (SBP-P). We also consider the DBE with discontinuous coefficients and present novel SBP-SAT, SBP-P and hybrid SBP-SAT-P discretizations for imposing interface conditions. Numerical experiments show that all methods considered are similar in terms of accuracy, but that SBP-P can be more computationally efficient (less restrictive time step requirement for explicit time integration methods) for both the constant and piecewise constant coefficient problems.