This report presents a comprehensive investigation into the accuracy and stability of Finite Difference Methods (FDM) when applied to the acoustic wave equation. The analysis focuses on comparing the classical 2nd order FDM with highly-accurate computational stencil of order 2p = 2,4 and 6 with Summation-by-Parts (SBP) and Simultaneous Approximation Term (SAT) technique of the Finite Difference Method. The objective of the study is to investigate complex numerical techniques that contributes to highly-accurate and stable solutions to many hyperbolic PDEs. The report starts by introducing the governing problem and studies its well-posedness to ensure stable and unique solutions of the governing equations. It continues with basic introduction to the classic spatial discretization of the FDM and introduces the SBP-SAT implementation of the method. The governing equations are rewritten as a semi-discrete problem, such that it can be written as a system of ordinary differential equations (ODEs) only dependent on the temporal evolution. This system can be solved with classic Runge-Kutta methods to ensure robust and accurate time-stepping schemes. The results show that the implementation of the higher-order SBP-SAT Finite Difference Method provides highly accurate solutions of the acoustic wave equation compared to the classic FDM. The results also show that the method provides stable solutions with no visible oscillations (dispersion), which can be a challenge for higher order methods. Overall, this paper contributes with valuable insights into the analysis of accuracy and stability in finite difference methods for acoustic wave equation.
| Identifer | oai:union.ndltd.org:UPSALLA1/oai:DiVA.org:kth-349408 |
| Date | January 2024 |
| Creators | Boughanmi, Aimen |
| Publisher | KTH, Skolan för teknikvetenskap (SCI) |
| Source Sets | DiVA Archive at Upsalla University |
| Language | English |
| Detected Language | English |
| Type | Student thesis, info:eu-repo/semantics/bachelorThesis, text |
| Format | application/pdf |
| Rights | info:eu-repo/semantics/openAccess |
| Relation | TRITA-SCI-GRU ; 2024:188 |
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