zbMATH — the first resource for mathematics

Hybridized summation-by-parts finite difference methods. (English) Zbl 07358934
Summary: We present a hybridization technique for summation-by-parts finite difference methods with weak enforcement of interface and boundary conditions for second order, linear elliptic partial differential equations. The method is based on techniques from the hybridized discontinuous Galerkin literature where local and global problems are defined for the volume and trace grid points, respectively. By using a Schur complement technique the volume points can be eliminated, which drastically reduces the system size. We derive both the local and global problems, and show that the resulting linear systems are symmetric positive definite. The theoretical stability results are confirmed with numerical experiments as is the accuracy of the method.
65N06 Finite difference methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J15 Second-order elliptic equations
65F05 Direct numerical methods for linear systems and matrix inversion
Julia ; CHOLMOD; CSparse
Full Text: DOI
[1] Bezanson, J.; Edelman, A.; Karpinski, S.; Shah, VB, Julia: a fresh approach to numerical computing, SIAM Rev., 59, 1, 65-98 (2017) · Zbl 1356.68030
[2] Carpenter, MH; Gottlieb, D.; Abarbanel, S., Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes, J. Comput. Phys., 111, 2, 220-236 (1994) · Zbl 0832.65098
[3] Carpenter, MH; Nordström, J.; Gottlieb, D., A stable and conservative interface treatment of arbitrary spatial accuracy, J. Comput. Phys., 148, 2, 341-365 (1999) · Zbl 0921.65059
[4] Chen, Y.; Davis, TA; Hager, WW; Rajamanickam, S., Algorithm 887: Cholmod, supernodal sparse Cholesky factorization and update/downdate, ACM Trans. Math. Softw., 3, 3, 22:1-22:14 (2008)
[5] Ciarlet, PG, The Finite Element Method for Elliptic Problems (2002), Philadelphia: Society for Industrial and Applied Mathematics, Philadelphia · Zbl 0999.65129
[6] Cockburn, B.; Gopalakrishnan, J.; Lazarov, R., Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal., 47, 2, 1319-1365 (2009) · Zbl 1205.65312
[7] Davis, TA, Direct methods for sparse linear systems (2006), Philadelphia: Society for Industrial and Applied Mathematics, Philadelphia · Zbl 1119.65021
[8] Erickson, BA; Day, SM, Bimaterial effects in an earthquake cycle model using rate-and-state friction, J. Geophys. Res. Solid Earth, 121, 2480-2506 (2016)
[9] Erickson, BA; Dunham, EM, An efficient numerical method for earthquake cycles in heterogeneous media: alternating subbasin and surface-rupturing events on faults crossing a sedimentary basin, J. Geophys. Res. Solid Earth, 119, 4, 3290-3316 (2014)
[10] Gassner, G., A skew-symmetric discontinuous Galerkin spectral element discretization and its relation to SBP-SAT finite difference methods, SIAM J. Sci. Comput., 35, 3, A1233-A1253 (2013) · Zbl 1275.65065
[11] George, A., Nested dissection of a regular finite element mesh, SIAM J. Numer. Anal., 10, 2, 345-363 (1973) · Zbl 0259.65087
[12] Guyan, RJ, Reduction of stiffness and mass matrices, AIAA J., 3, 2, 380 (1965)
[13] Karlstrom, L.; Dunham, EM, Excitation and resonance of acoustic-gravity waves in a column of stratified, bubbly magma, J. Fluid Mech., 797, 431-470 (2016) · Zbl 1422.86016
[14] Kozdon, JE; Dunham, M.; Nordström, J., Interaction of waves with frictional interfaces using summation-by-parts difference operators: weak enforcement of nonlinear boundary conditions, J. Sci. Comput., 50, 341-367 (2012) · Zbl 1325.74161
[15] Kozdon, JE; Wilcox, LC, Stable coupling of nonconforming, high-order finite difference methods, SIAM J. Sci. Comput., 38, 2, A923-A952 (2016) · Zbl 1380.65160
[16] Kreiss, H., Scherer, G.: Finite element and finite difference methods for hyperbolic partial differential equations. In: Mathematical Aspects of Finite Elements in Partial Differential Equations; Proceedings of the Symposium, Madison, WI, pp. 195-212 (1974). doi:10.1016/b978-0-12-208350-1.50012-1 · Zbl 0355.65085
[17] Kreiss, H., Scherer, G.: On the Existence of Energy Estimates for Difference Approximations for Hyperbolic Systems. Technical report, Department of Scientific Computing, Uppsala University (1977)
[18] Lotto, GC; Dunham, EM, High-order finite difference modeling of tsunami generation in a compressible ocean from offshore earthquakes, Comput. Geosci., 19, 2, 327-340 (2015) · Zbl 1392.86016
[19] Mattsson, K., Summation by parts operators for finite difference approximations of second-derivatives with variable coefficients, J. Sci. Comput., 51, 3, 650-682 (2012) · Zbl 1252.65055
[20] Mattsson, K.; Carpenter, MH, Stable and accurate interpolation operators for high-order multiblock finite difference methods, SIAM J. Sci. Comput., 32, 4, 2298-2320 (2010) · Zbl 1216.65107
[21] Mattsson, K.; Ham, F.; Iaccarino, G., Stable boundary treatment for the wave equation on second-order form, J. Sci. Comput., 41, 3, 366-383 (2009) · Zbl 1203.65145
[22] Mattsson, K.; Nordström, J., Summation by parts operators for finite difference approximations of second derivatives, J. Comput. Phys., 199, 2, 503-540 (2004) · Zbl 1071.65025
[23] Mattsson, K.; Parisi, F., Stable and accurate second-order formulation of the shifted wave equation, Commun. Comput. Phys., 7, 1, 103 (2010) · Zbl 1364.65167
[24] Nissen, A.; Kreiss, G.; Gerritsen, M., High order stable finite difference methods for the Schrödinger equation, J. Sci. Comput., 55, 1, 173-199 (2013) · Zbl 1273.65112
[25] Nordström, J.; Carpenter, MH, High-order finite difference methods, multidimensional linear problems, and curvilinear coordinates, J. Comput. Phys., 173, 1, 149-174 (2001) · Zbl 0987.65081
[26] Roache, P., Verification and Validation in Computational Science and Engineering (1998), Albuquerque: Hermosa Publishers, Albuquerque
[27] Ruggiu, AA; Weinerfelt, P.; Nordström, J., A new multigrid formulation for high order finite difference methods on summation-by-parts form, J. Comput. Phys., 359, 216-238 (2018) · Zbl 1422.65442
[28] Strand, B., Summation by parts for finite difference approximations for \(d/dx\), J. Comput. Phys., 110, 1, 47-67 (1994) · Zbl 0792.65011
[29] Thomée, V.: From finite differences to finite elements: a short history of numerical analysis of partial differential equations. In: Numerical Analysis: Historical Developments in the 20th Century, pp. 361-414. Elsevier, Amsterdam (2001). doi:10.1016/S0377-0427(00)00507-0 · Zbl 0977.65001
[30] Virta, K.; Mattsson, K., Acoustic wave propagation in complicated geometries and heterogeneous media, J. Sci. Comput., 61, 1, 90-118 (2014) · Zbl 1306.65247
[31] Wang, S.; Virta, K.; Kreiss, G., High order finite difference methods for the wave equation with non-conforming grid interfaces, J. Sci. Comput., 68, 3, 1002-1028 (2016) · Zbl 1352.65274
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.