2020/01/01 by Alexander Zlotnik, Zlotnik, Alexander, Raimondas Čiegis +1
Engineering · Mathematics · #65M06 #65M12 #Advanced Mathematical Physics Problems #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2012.01000
openalex publication_date 2020/01/01 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
We study necessary conditions for stability of a Numerov-type compact\nhigher-order finite-difference scheme for the 1D homogeneous wave equation in\nthe case of non-uniform spatial meshes. We first show that the uniform in time\nstability cannot be valid in any spatial norm provided that the complex\neigenvalues appear in the associated mesh eigenvalue problem. Moreover, we\nprove that then the solution norm grows exponentially in time making the scheme\nstrongly non-dissipative and therefore impractical. Numerical results confirm\nthis conclusion. In addition, for some sequences of refining spatial meshes, an\nexcessively strong condition between steps in time and space is necessary (even\nfor the non-uniform in time stability) which is familiar for explicit schemes\nin the parabolic case.\n