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Summation-by-Parts Finite Difference Methods for Wave Propagation and Earthquake Modeling
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing.ORCID iD: 0000-0003-4264-3234
2023 (English)Doctoral thesis, comprehensive summary (Other academic)
Description
Abstract [en]

Waves manifest in many areas of physics, ranging from large-scale seismic waves in geophysics down to particle descriptions in quantum physics. Wave propagation may often be described mathematically by partial differential equations (PDE). Unfortunately, analytical solutions to PDEs are in many cases notoriously difficult to obtain. For this reason, one turns to approximate solutions obtained through numerical methods implemented as computer algorithms. In order for a numerical method to be useful in predictive simulations, it should be stable and accurate. Stability of the method ensures that small errors in the approximation do not grow exponentially. Accuracy together with stability ensures that increased resolution in the simulation results in decreased error in the approximation. The numerical methods considered in this thesis are finite difference methods satisfying a summation-by-parts (SBP) property. Finite difference methods are well suited for wave propagation problems in that they provide high accuracy at low computational cost. The SBP property additionally facilitates the construction of provably stable high-order accurate approximations.

This thesis continues the development of SBP finite difference methods for wave propagation problems. Paper I presents a finite difference method for modeling induced seismicity, i.e., earthquakes caused by human activity. Paper II develops a high-order accurate finite difference method for shock waves modeled by scalar conservation laws. In Paper III, new SBP finite difference operators with increased accuracy and efficiency for surface and interface waves are derived. In Papers IV - V numerical methods for inverse problems governed by wave equations are considered, where unknown model parameters are reconstructed by fitting the numerical solution to known data. Specifically, Paper IV presents a method for acoustic shape optimization, while Paper V presents an inversion method for frictional parameters used in earthquake modeling.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2023. , p. 50
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 2327
Keywords [en]
Finite difference method, high-order accuracy, stability, summation-by-parts, wave propagation, earthquake modeling, inverse problems
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
URN: urn:nbn:se:uu:diva-514589ISBN: 978-91-513-1936-0 (print)OAI: oai:DiVA.org:uu-514589DiVA, id: diva2:1805903
Public defence
2023-12-08, Sonja Lyttkens, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 10:15 (English)
Opponent
Supervisors
Funder
Swedish Research Council, 2017-04626Available from: 2023-11-13 Created: 2023-10-18 Last updated: 2023-11-13
List of papers
1. A finite difference method for earthquake sequences in poroelastic solids
Open this publication in new window or tab >>A finite difference method for earthquake sequences in poroelastic solids
2018 (English)In: Computational Geosciences, ISSN 1420-0597, E-ISSN 1573-1499, Vol. 22, p. 1351-1370Article in journal (Refereed) Published
National Category
Computational Mathematics Geophysics
Identifiers
urn:nbn:se:uu:diva-356436 (URN)10.1007/s10596-018-9757-1 (DOI)000444706400012 ()
Available from: 2018-07-19 Created: 2018-07-27 Last updated: 2023-10-18Bibliographically approved
2. A residual-based artificial viscosity finite difference method for scalar conservation laws
Open this publication in new window or tab >>A residual-based artificial viscosity finite difference method for scalar conservation laws
2021 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 430, article id 110100Article in journal (Refereed) Published
Abstract [en]

In this paper, we present an accurate, stable and robust shock-capturing finite difference method for solving scalar non-linear conservation laws. The spatial discretization uses high-order accurate upwind summation-by-parts finite difference operators combined with weakly imposed boundary conditions via simultaneous-approximation-terms. The method is an extension of the residual-based artificial viscosity methods developed in the finite- and spectral element communities to the finite difference setting. The three main ingredients of the proposed method are: (i) shock detection provided by a residual-based error estimator; (ii) first-order viscosity applied in regions with strong discontinuities; (iii) additional dampening of spurious oscillations provided by high-order dissipation from the upwind finite difference operators. The method is shown to be stable for skew-symmetric discretizations of the advective flux. Accuracy and robustness are shown by solving several benchmark problems in 2D for convex and non-convex fluxes.

Place, publisher, year, edition, pages
ElsevierACADEMIC PRESS INC ELSEVIER SCIENCE, 2021
Keywords
High-order finite difference methods, Conservation laws, Shock-capturing, Artificial viscosity, Residual-based error estimator, SBP-SAT
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-441161 (URN)10.1016/j.jcp.2020.110100 (DOI)000624309300010 ()
Funder
eSSENCE - An eScience Collaboration
Available from: 2021-05-06 Created: 2021-05-06 Last updated: 2024-01-15Bibliographically approved
3. Boundary-optimized summation-by-parts operators for finite difference approximations of second derivatives with variable coefficients
Open this publication in new window or tab >>Boundary-optimized summation-by-parts operators for finite difference approximations of second derivatives with variable coefficients
2023 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 491, article id 112376Article in journal (Refereed) Epub ahead of print
Abstract [en]

Boundary-optimized summation-by-parts (SBP) finite difference operators for second derivatives with variable coefficients are presented. The operators achieve increased accuracy by utilizing non-equispaced grid points close to the boundaries of the grid. Using the optimized operators we formulate SBP schemes for the acoustic and elastic operators defined directly on curvilinear multiblock domains. Numerical studies of the acoustic and elastic wave equations demonstrate that, compared to traditional SBP difference operators, the new operators provide increased accuracy for surface waves as well as block interfaces in multiblock grids. For instance, simulations of Rayleigh waves demonstrate that the boundary-optimized operators more than halve the runtime required for a given error tolerance.

Keywords
High-order finite difference methods, Second derivative, Summation by parts, Boundary accuracy, Wave propagation, Surface waves
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
urn:nbn:se:uu:diva-509841 (URN)10.1016/j.jcp.2023.112376 (DOI)
Available from: 2023-08-23 Created: 2023-08-23 Last updated: 2023-10-18Bibliographically approved
4. Acoustic shape optimization using energy stable curvilinear finite differences
Open this publication in new window or tab >>Acoustic shape optimization using energy stable curvilinear finite differences
(English)Manuscript (preprint) (Other academic)
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-514581 (URN)10.48550/arXiv.2310.11956 (DOI)
Funder
Swedish Research Council Formas, 2018-00925Swedish Research Council, 2017-04626 VR
Available from: 2023-10-18 Created: 2023-10-18 Last updated: 2023-11-12
5. Adjoint-based inversion for stress and frictional parameters in earthquake modeling
Open this publication in new window or tab >>Adjoint-based inversion for stress and frictional parameters in earthquake modeling
(English)Manuscript (preprint) (Other academic)
National Category
Computational Mathematics Geophysics
Identifiers
urn:nbn:se:uu:diva-514582 (URN)10.48550/arXiv.2310.12279 (DOI)
Funder
Swedish Research Council, 2017-0462
Available from: 2023-10-18 Created: 2023-10-18 Last updated: 2023-11-12

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