Inverses of SBP-SAT Finite Difference Operators Approximating the First and Second Derivative

Abstract

The scalar, one-dimensional advection equation and heat equation are considered. These equations are discretized in space, using a finite difference method satisfying summation-by-parts (SBP) properties. To impose the boundary conditions, we use a penalty method called simultaneous approximation term (SAT). Together, this gives rise to two semi-discrete schemes where the discretization matrices approximate the first and the second derivative operators, respectively. The discretization matrices depend on free parameters from the SAT treatment. We derive the inverses of the discretization matrices, interpreting them as discrete Green’s functions. In this direct way, we also find out precisely which choices of SAT parameters that make the discretization matrices singular. In the second derivative case, it is shown that if the penalty parameters are chosen such that the semi-discrete scheme is dual consistent, the discretization matrix can become singular even when the scheme is energy stable. The inverse formulas hold for SBP-SAT operators of arbitrary order of accuracy. For second and fourth order accurate operators, the inverses are provided explicitly.

An Energy-Based Summation-by-Parts Finite Difference Method For the Wave Equation in Second Order Form

“…Since λ 1 = 0, the rank of A is n − 1, and Assumption 2 is true. For the second and fourth order SBP operators with constant coefficient, Assumption 2 is proved in [10] using the result from [9], and the explicit formulas for the Moore-Penrose inverse of A are also derived.…”

“…The auxiliary variables μ and ν satisfy μ t ≈ ν. Alternatively, we can use the pseudoinverse of A (b) . Since the right-hand side of ( 9) is a summation of rank-one vectors, we only need a few columns of the pseudoinverse of A (b) , which can be computed by using the analytical formula in [9,10] for constant coefficient problems and p = 2 or 4.…”