The numerical solution of semidiscrete linear evolution problems on the finite interval using the Unified Transform Method
Authors:
Jorge Cisneros and Bernard Deconinck
Journal:
Quart. Appl. Math. 80 (2022), 739-786
MSC (2020):
Primary 65M22, 65M06; Secondary 39A27, 39A14
DOI:
https://doi.org/10.1090/qam/1626
Published electronically:
June 29, 2022
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Abstract: We study a semidiscrete analogue of the Unified Transform Method introduced by A. S. Fokas, to solve initial-boundary-value problems for linear evolution partial differential equations with constant coefficients on the finite interval $x \in (0,L)$. The semidiscrete method is applied to various spatial discretizations of several first and second-order linear equations, producing the exact solution for the semidiscrete problem, given appropriate initial and boundary data. From these solutions, we derive alternative series representations that are better suited for numerical computations. In addition, we show how the Unified Transform Method treats derivative boundary conditions and ghost points introduced by the choice of discretization stencil and we propose the notion of “natural” discretizations. We consider the continuum limit of the semidiscrete solutions and compare with standard finite-difference schemes.
References
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- Lloyd N. Trefethen and Mark Embree, Spectra and pseudospectra, Princeton University Press, Princeton, NJ, 2005. The behavior of nonnormal matrices and operators. MR 2155029, DOI 10.1515/9780691213101
- R. F. Warming and B. J. Hyett, The modified equation approach to the stability and accuracy analysis of finite-difference methods, J. Comput. Phys. 14 (1974), 159–179. MR 339526, DOI 10.1016/0021-9991(74)90011-4
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References
- Gino Biondini and Guenbo Hwang, Initial-boundary-value problems for discrete evolution equations: discrete linear Schrödinger and integrable discrete nonlinear Schrödinger equations, Inverse Problems 24 (2008), no. 6, 065011, 44. MR 2456958, DOI 10.1088/0266-5611/24/6/065011
- Gino Biondini and Danhua Wang, Initial-boundary-value problems for discrete linear evolution equations, IMA J. Appl. Math. 75 (2010), no. 6, 968–997. MR 2740041, DOI 10.1093/imamat/hxq014
- D. Britz, R. Baronas, E. Gaidamauskaitė, and F. Ivanauskas, Further comparisons of finite difference schemes for computational modelling of biosensors, Nonlinear Anal. Model. Control 14 (2009), no. 4, 419–433. MR 2603686
- T. A. Cheema, Higher-order finite-difference methods for partial differential equations, PhD Thesis, Brunel University, Middlesex, England (1997).
- J. Cisneros and B. Deconinck, The numerical solutions of linear semidiscrete evolution problems on the half-line using the unified transform method, Studies in Applied Mathematics 147 (2021), no. 4, 1240–1276.
- J. Cisneros and B. Deconinck, The numerical solutions of third-order linear semi-discrete evolution problems using the unified transform method, In preparation (2021).
- Armando Coco and Giovanni Russo, Finite-difference ghost-point multigrid methods on Cartesian grids for elliptic problems in arbitrary domains, J. Comput. Phys. 241 (2013), 464–501. MR 3647417, DOI 10.1016/j.jcp.2012.11.047
- Matthew J. Colbrook, Zdravko I. Botev, Karsten Kuritz, and Shev MacNamara, Kernel density estimation with linked boundary conditions, Stud. Appl. Math. 145 (2020), no. 3, 357–396. MR 4174161, DOI 10.1111/sapm.12322
- Bernard Deconinck, Thomas Trogdon, and Vishal Vasan, The method of Fokas for solving linear partial differential equations, SIAM Rev. 56 (2014), no. 1, 159–186. MR 3246302, DOI 10.1137/110821871
- Bernard Deconinck, Thomas Trogdon, and Xin Yang, The numerical unified transform method for initial-boundary value problems on the half-line, IMA J. Numer. Anal. 42 (2022), no. 2, 1400–1433. MR 4410746, DOI 10.1093/imanum/drab007
- Daniel J. Duffy, Finite difference methods in financial engineering, Wiley Finance Series, John Wiley & Sons, Ltd., Chichester, 2006. A partial differential equation approach; With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2286409, DOI 10.1002/9781118673447
- Dale R. Durran, Numerical methods for fluid dynamics, 2nd ed., Texts in Applied Mathematics, vol. 32, Springer, New York, 2010. With applications to geophysics. MR 2723959, DOI 10.1007/978-1-4419-6412-0
- N. Flyer and A. S. Fokas, A hybrid analytical-numerical method for solving evolution partial differential equations. I. The half-line, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 464 (2008), no. 2095, 1823–1849. MR 2403130, DOI 10.1098/rspa.2008.0041
- A. S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London Ser. A 453 (1997), no. 1962, 1411–1443. MR 1469927, DOI 10.1098/rspa.1997.0077
- Athanassios S. Fokas, A unified approach to boundary value problems, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 78, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. MR 2451953, DOI 10.1137/1.9780898717068
- A. S. Fokas and B. Pelloni, Unified transform for boundary value problems, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2015, Applications and advances. MR 3364228
- E. Gaidamauskaitė and R. Baronas, A comparison of finite difference schemes for computational modelling of biosensors, Nonlinear Analysis: Modelling and Control 12 (2007), no. 3, 359–369.
- K. George and E. H. Twizell, Stable second-order finite-difference methods for linear initial-boundary-value problems, Appl. Math. Lett. 19 (2006), no. 2, 146–154. MR 2198401, DOI 10.1016/j.aml.2005.04.003
- Bertil Gustafsson, Heinz-Otto Kreiss, and Arne Sundström, Stability theory of difference approximations for mixed initial boundary value problems. II, Math. Comp. 26 (1972), 649–686. MR 341888, DOI 10.2307/2005093
- Arieh Iserles, A first course in the numerical analysis of differential equations, 2nd ed., Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2009. MR 2478556
- Emine Kesici, Beatrice Pelloni, Tristan Pryer, and David Smith, A numerical implementation of the unified Fokas transform for evolution problems on a finite interval, European J. Appl. Math. 29 (2018), no. 3, 543–567. MR 3788455, DOI 10.1017/S0956792517000316
- Hans Petter Langtangen and Svein Linge, Finite difference computing with PDEs, Texts in Computational Science and Engineering, vol. 16, Springer, Cham, 2017. A modern software approach. MR 3675518, DOI 10.1007/978-3-319-55456-3
- Randall J. LeVeque, Intermediate boundary conditions for time-split methods applied to hyperbolic partial differential equations, Math. Comp. 47 (1986), no. 175, 37–54. MR 842122, DOI 10.2307/2008081
- R. J. LeVeque, Finite difference methods for ordinary and partial differential equations, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2007, Steady-state and time-dependent problems. MR 2378550
- Randall J. LeVeque and Joseph Oliger, Numerical methods based on additive splittings for hyperbolic partial differential equations, Math. Comp. 40 (1983), no. 162, 469–497. MR 689466, DOI 10.2307/2007526
- Shev MacNamara and Gilbert Strang, Operator splitting, Splitting methods in communication, imaging, science, and engineering, Sci. Comput., Springer, Cham, 2016, pp. 95–114. MR 3617561
- Byungsoo Moon and Guenbo Hwang, Discrete linear evolution equations in a finite lattice, J. Difference Equ. Appl. 25 (2019), no. 5, 630–646. MR 3977231, DOI 10.1080/10236198.2019.1613386
- Theodore S. Papatheodorou and Anastasia N. Kandili, Novel numerical techniques based on Fokas transforms, for the solution of initial boundary value problems, J. Comput. Appl. Math. 227 (2009), no. 1, 75–82. MR 2512761, DOI 10.1016/j.cam.2008.07.031
- Chaopeng Shen, Jing-Mei Qiu, and Andrew Christlieb, Adaptive mesh refinement based on high order finite difference WENO scheme for multi-scale simulations, J. Comput. Phys. 230 (2011), no. 10, 3780–3802. MR 2783818, DOI 10.1016/j.jcp.2011.02.008
- John C. Strikwerda, Finite difference schemes and partial differential equations, 2nd ed., Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2004. MR 2124284, DOI 10.1137/1.9780898717938
- Surattana Sungnul, Bubpha Jitsom, and Mahosut Punpocha, Numerical solutions of the modified Burger’s equation using FTCS implicit scheme, IAENG Int. J. Appl. Math. 48 (2018), no. 1, 53–61. MR 3791243
- Béla Szilágyi, Denis Pollney, Luciano Rezzolla, Jonathan Thornburg, and Jeffrey Winicour, An explicit harmonic code for black-hole evolution using excision, Classical Quantum Gravity 24 (2007), no. 12, S275–S293. MR 2333164, DOI 10.1088/0264-9381/24/12/S18
- Lloyd N. Trefethen and Mark Embree, Spectra and pseudospectra, Princeton University Press, Princeton, NJ, 2005. The behavior of nonnormal matrices and operators. MR 2155029
- R. F. Warming and B. J. Hyett, The modified equation approach to the stability and accuracy analysis of finite-difference methods, J. Comput. Phys. 14 (1974), 159–179. MR 339526, DOI 10.1016/0021-9991(74)90011-4
- Daniel Zwillinger, CRC standard mathematical tables and formulae, 2nd ed., CRC Press, Boca Raton, FL, 2012. MR 2933767
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Additional Information
Jorge Cisneros
Affiliation:
Department of Applied Mathematics, University of Washington, Seattle, Washington 98195-2420
MR Author ID:
1242436
ORCID:
0000-0003-4493-0536
Email:
jorgec5@uw.edu
Bernard Deconinck
Affiliation:
Department of Applied Mathematics, University of Washington, Seattle, Washington 98195-2420
MR Author ID:
613566
Email:
bernard@amath.washington.edu
Keywords:
Continuum limit,
finite difference,
finite interval,
ghost points,
semidiscrete linear problem,
Unified Transform Method
Received by editor(s):
February 4, 2022
Received by editor(s) in revised form:
April 24, 2022
Published electronically:
June 29, 2022
Additional Notes:
This work was supported by the Graduate Opportunities & Minority Achievement Program Fellowship from the University of Washington and the Ford Foundation Predoctoral Fellowship (the first author). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the funding sources.
Article copyright:
© Copyright 2022
Brown University
