On compact 4th order finite-difference schemes for the wave equation

A compact higher-order finite-difference scheme for the wave equation can be strongly non-dissipative on non-uniform meshes

“…In this paper, we deal with a Numerov-type compact scheme for the 1D homogeneous wave equation. Its uniform in time conditional stability and higher-order error properties even for non-smooth data have recently been demonstrated in [9] in the case of uniform meshes (and the non-homogeneous wave equation). However, in the case of non-uniform spatial meshes, the situation can change dramatically.…”

“…for several its derivations and equivalent forms of α h , β h and γ h , see [3,5,7,9]. One can check easily that 1 12 (α h + 10γ h + β h ) = 1 on ω h .…”

“…Vast literature is devoted to compact higher-order finite-difference schemes for various PDEs; we call only some recent papers for the wave equation [1,4,6,9]; see also references therein. In the case of uniform meshes and PDEs with constant coefficients, their stability properties are often similar to more standard schemes and can be derived in a similar manner too.…”

“…Recall that 2 3 w ω h w s N w ω h for any w ∈ H h . It follows from the proof in [9] and [10, Lemma 2.1] that the left-hand side of ( 8) can be replaced by c(ε 0 )a 2 ( v m 2 −Λ + v m+1 2 −Λ ) with some c(ε 0 ) > 0. These bounds imply similar ones for the complex v 0 and u 1N as well.…”

“…For the uniform mesh ωh with the step h, we have s N w = w + h 2 12 Λw and thus the eigenvalues λ are positive real and are found explicitly; in particular, the maximal of them equals [9]…”