Journal
APPLIED MATHEMATICS LETTERS
Volume 115, Issue -, Pages -Publisher
PERGAMON-ELSEVIER SCIENCE LTD
DOI: 10.1016/j.aml.2020.106949
Keywords
Wave equation; Compact higher-order; finite-difference scheme; Non-uniform mesh; Stability
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Funding
- Academic Fund Program at the National Research University Higher School of Economics (HSE), Russia in 2019-2020 [19-01-021]
- Russian Foundation for the Basic Research [19-01-00262]
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The study examines the necessary conditions for stability of a Numerov-type compact higher-order finite-difference scheme for the 1D homogeneous wave equation on non-uniform spatial meshes. It is found that exponential growth in solution norm and excessively strong conditions between time and space steps are required for stability, even in the case of non-uniform time stability.
We study necessary conditions for stability of a Numerov-type compact higher-order finite-difference scheme for the 1D homogeneous wave equation in the case of non-uniform spatial meshes. We first show that the uniform in time stability cannot be valid in any spatial norm provided that the complex eigenvalues appear in the associated mesh eigenvalue problem. Moreover, we prove that then the solution norm grows exponentially in time making the scheme strongly non-dissipative and therefore impractical. Numerical results confirm this conclusion. In addition, for some sequences of refining spatial meshes, an excessively strong condition between steps in time and space is necessary (even for the non-uniform in time stability) which is familiar for explicit schemes in the parabolic case. (C) 2020 Elsevier Ltd. All rights reserved.
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