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Energy conservation issues in the numerical solution of the semilinear wave equation. (English) Zbl 1410.65477

Summary: In this paper we discuss energy conservation issues related to the numerical solution of the semilinear wave equation. As is well known, this problem can be cast as a Hamiltonian system that may be autonomous or not, depending on the prescribed boundary conditions. We relate the conservation properties of the original problem to those of its semi-discrete version obtained by the method of lines. Subsequently, we show that the very same properties can be transferred to the solutions of the fully discretized problem, obtained by using energy-conserving methods in the HBVMs (Hamiltonian Boundary Value Methods) class. Similar arguments hold true for different types of Hamiltonian partial differential equations, e.g., the nonlinear Schrödinger equation.

MSC:

65P10 Numerical methods for Hamiltonian systems including symplectic integrators
35L20 Initial-boundary value problems for second-order hyperbolic equations
35L72 Second-order quasilinear hyperbolic equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs

Software:

BiMD; Mulprec; BiM
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References:

[1] P. Amodio, L. Brugnano, F. Iavernaro, Energy-conserving methods for Hamiltonian boundary value problems and applications in astrodynamics, Adv. Comput. Math. doi:10.1007/s10444-014-9390-z. · Zbl 1327.65267
[2] Amodio, P.; Sgura, I., High-order finite difference schemes for the solution of second-order BVPs, J. Comput. Appl. Math., 176, 59-76, (2005) · Zbl 1073.65061
[3] Betsch, P.; Steinmann, P., Inherently energy conserving time finite elements for classical mechanics, J. Comput. Phys., 160, 1, 88-116, (2000) · Zbl 0966.70003
[4] Betsch, P.; Steinmann, P., Conservation properties of a time FE method. I. time-stepping schemes for N-body problems, Int. J. Numer. Methods Eng., 49, 5, 599-638, (2000) · Zbl 0964.70002
[5] Boyd, J. P., Chebyshev and Fourier Spectral Methods, (2001), Dover Publications Inc. Mineola, NY · Zbl 0994.65128
[6] Bridges, T. J., Multisymplectic structures and wave propagation, Math. Proc. Camb. Philos. Soc., 121, 147-190, (1997) · Zbl 0892.35123
[7] Bridges, T. J.; Reich, S., Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity, Phys. Lett. A, 284, 184-193, (2001) · Zbl 0984.37104
[8] Bridges, T. J.; Reich, S., Multi-symplectic spectral discretizations for the Zakharov-Kuznetsov and shallow water equations, Physica D, 152, 491-504, (2001) · Zbl 1032.76053
[9] Bridges, T. J.; Reich, S., Numerical methods for Hamiltonian pdes, J. Phys. A: Math. Gen., 39, 5287-5320, (2006) · Zbl 1090.65138
[10] Brugnano, L., Blended block BVMs (B3VMs): a family of economical implicit methods for odes, J. Comput. Appl. Math., 116, 41-62, (2000) · Zbl 0982.65084
[11] Brugnano, L.; Calvo, M.; Montijano, J. I.; Ràndez, L., Energy preserving methods for Poisson systems, J. Comput. Appl. Math., 236, 3890-3904, (2012) · Zbl 1247.65092
[12] Brugnano, L.; Frasca Caccia, G.; Iavernaro, F., Efficient implementation of Gauss collocation and Hamiltonian boundary value methods, Numer. Algor., 65, 633-650, (2014) · Zbl 1291.65357
[13] Brugnano, L.; Frasca Caccia, G.; Iavernaro, F., Efficient implementation of geometric integrators for separable Hamiltonian problems, AIP Conf. Proc., 1558, 734-737, (2013)
[14] Brugnano, L.; Iavernaro, F., Line integral methods which preserve all invariants of conservative problems, J. Comput. Appl. Math., 236, 3905-3919, (2012) · Zbl 1246.65108
[15] Brugnano, L.; Iavernaro, F.; Trigiante, D., Analysis of Hamiltonian boundary value methods (HBVMs): a class of energy-preserving Runge-Kutta methods for the numerical solution of polynomial Hamiltonian systems, Commun. Nonlin. Sci. Numer. Simul., 20, 650-667, (2015) · Zbl 1304.65262
[16] Brugnano, L.; Iavernaro, F.; Trigiante, D., Hamiltonian BVMs (HBVMs): a family of “drift-free” methods for integrating polynomial Hamiltonian systems, AIP Conf. Proc., 1168, 715-718, (2009) · Zbl 1182.65188
[17] Brugnano, L.; Iavernaro, F.; Trigiante, D., Hamiltonian boundary value methods (energy preserving discrete line methods), J. Numer. Anal. Ind. Appl. Math., 5, 1-2, 17-37, (2010) · Zbl 1432.65182
[18] Brugnano, L.; Iavernaro, F.; Trigiante, D., A note on the efficient implementation of Hamiltonian bvms, J. Comput. Appl. Math., 236, 375-383, (2011) · Zbl 1228.65107
[19] Brugnano, L.; Iavernaro, F.; Trigiante, D., The lack of continuity and the role of infinite and infinitesimal in numerical methods for ODEs: the case of symplecticity, Appl. Math. Comput., 218, 8053-8063, (2012) · Zbl 1245.65085
[20] Brugnano, L.; Iavernaro, F.; Trigiante, D., A simple framework for the derivation and analysis of effective one-step methods for odes, Appl. Math. Comput., 218, 8475-8485, (2012) · Zbl 1245.65086
[21] Brugnano, L.; Iavernaro, F.; Trigiante, D., A two-step, fourth-order method with energy preserving properties, Comput. Phys. Commun., 183, 1860-1868, (2012) · Zbl 1305.65238
[22] Brugnano, L.; Iavernaro, F.; Trigiante, D., Energy and quadratic invariants preserving integrators based upon Gauss collocation formulae, SIAM J. Numer. Anal., 50, 6, 2897-2916, (2012) · Zbl 1261.65130
[23] Brugnano, L.; Magherini, C., Blended implementation of block implicit methods for odes, Appl. Numer. Math., 42, 29-45, (2002) · Zbl 1006.65078
[24] Brugnano, L.; Magherini, C., The BIM code for the numerical solution of ODES, J. Comput. Appl. Math., 164-165, 145-158, (2004) · Zbl 1038.65063
[25] Brugnano, L.; Sun, Y., Multiple invariants conserving Runge-Kutta type methods for Hamiltonian problems, Numer. Algor., 65, 611-632, (2014) · Zbl 1291.65358
[26] Cano, B., Conserved quantities of some Hamiltonian wave equations after full discretization, Numer. Math., 103, 197-223, (2006) · Zbl 1096.65125
[27] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral Methods in Fluid Dynamics, (1988), Springer-Verlag New York · Zbl 0658.76001
[28] Carpenter, M.; Gottlieb, D.; Abarbanel, S., Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes, J. Comp. Phys., 111, 220-236, (1994) · Zbl 0832.65098
[29] Celledoni, E.; Grimm, V.; McLachlan, R. I.; McLaren, D. I.; O’Neale, D.; Owren, B.; Quispel, G. R.W., Preserving energy resp. dissipation in numerical PDEs using the “average vector field” method, J. Comput. Phys., 231, 20, 6770-6789, (2012) · Zbl 1284.65184
[30] Celledoni, E.; McLachlan, R. I.; McLaren, D. I.; Owren, B.; Quispel, G. R.W.; Wright, W. M., Energy-preserving Runge-Kutta methods, M2AN Math. Model. Numer. Anal., 43, 4, 645-649, (2009) · Zbl 1169.65348
[31] Celledoni, E.; Owren, B.; Sun, Y., The minimal stage, energy preserving Runge-Kutta method for polynomial Hamiltonian systems is the averaged vector field method, Math. Comp., 83, 288, 1689-1700, (2014) · Zbl 1296.65182
[32] Chen, J. B.; Qin, M. Z., Multi-symplectic Fourier pseudospectral method for the nonlinear Schrödinger equation, Electron. Trans. Numer. Anal., 12, 193-204, (2001) · Zbl 0980.65108
[33] Cohen, D.; Hairer, E.; Lubich, C., Conservation of energy, momentum and actions in numerical discretizations of non-linear wave equations, Numer. Math., 110, 113-143, (2008) · Zbl 1163.65066
[34] Dahlquist, G.; Bijörk, Å., Numerical Methods in Scientific Computing, vol. 1, (2008), SIAM Philadelphia
[35] Evans, G. A.; Webster, J. R., A comparison of some methods for the evaluation of highly oscillatory integrals, J. Comput. Appl. Math., 112, 55-69, (1999) · Zbl 0947.65148
[36] Evans, L. C., Partial Differential Equations, (2010), AMS
[37] Faou, E., Geometric Numerical Integration and Schrödinger Equations, Zurich Lectures in Advanced Mathematics, (2012), European Mathematical Society (EMS) Zürich
[38] Flå, T., A numerical energy conserving method for the DNLS equation, J. Comput. Phys., 101, 71-79, (1992) · Zbl 0768.65079
[39] Forneberg, B.; Whitham, G. B., A numerical and theoretical study of certain nonlinear wave phenomena, Proc. R. Soc. Lond. A, 289, 373-403, (1978) · Zbl 0384.65049
[40] Frank, J., Conservation of wave action under multisymplectic discretizations, J. Phys. A: Math. Gen., 39, 5479-5493, (2006) · Zbl 1095.65112
[41] Frank, J.; Moore, B. E.; Reich, S., Linear PDEs and numerical methods that preserve a multisymplectic conservation law, SIAM J. Sci. Comput., 28, 260-277, (2006) · Zbl 1113.65117
[42] de Frutos, J.; Ortega, T.; Sanz-Serna, J. M., A Hamiltonian, explicit algorithm with spectral accuracy for the “good” Boussinesq system, Comput. Methods Appl. Mech. Eng., 80, 417-423, (1990) · Zbl 0728.76072
[43] Funaro, D.; Gottlieb, D., A new method of imposing boundary conditions for hyperbolic equations, Math. Comp., 51, 599-613, (1988) · Zbl 0699.65079
[44] Furihata, D., Finite-difference schemes for nonlinear wave equation that inherit energy conservation property, J. Comput. Appl. Math., 134, 1-2, 37-57, (2001) · Zbl 0989.65099
[45] Furihata, D.; Matsuo, T., Discrete Variational Derivative Method. A Structure-preserving Numerical Method for Partial Differential Equations, (2011), CRC Press Boca Raton, FL · Zbl 1227.65094
[46] Gustafsson, B., High Order Difference Methods for Time Dependent PDE, (2008), Springer-Verlag Berlin · Zbl 1146.65064
[47] Hairer, E., Energy-preserving variant of collocation methods, JNAIAM J. Numer. Anal. Ind. Appl. Math., 5, 1-2, 73-84, (2010) · Zbl 1432.65185
[48] Huang, M., A Hamiltonian approximation to simulate solitary waves of the kortweg-de Vries equation, Math. Comp., 56, 194, 607-620, (1991) · Zbl 0723.65100
[49] Hairer, E.; Lubich, C.; Wanner, G., Geometric Numerical Integration. Structure-preserving Algorithms for Ordinary Differential Equations, (2006), Springer-Verlag Berlin · Zbl 1094.65125
[50] Hairer, E.; Lubich, C., Spectral semi-discretisations of weakly nonlinear wave equations over long times, Found. Comput. Math., 8, 319-334, (2008) · Zbl 1156.35007
[51] Herbst, B. M.; Ablowitz, M. J., Numerical chaos, symplectic integrators, and exponentially small splitting distances, J. Comput. Phys., 105, 1, 122-132, (1993) · Zbl 0772.65084
[52] Hu, W.; Deng, Z.; Han, S.; Zhang, W., Generalized multi-symplectic integrators for a class of Hamiltonian nonlinear wave pdes, J. Comput. Phys., 235, 394-406, (2013) · Zbl 1291.65361
[53] Koide, S.; Furihata, D., Nonlinear and linear conservative finite difference schemes for regularized long wave equation, Jpn. J. Ind. Appl. Math., 26, 1, 15-40, (2009) · Zbl 1177.65124
[54] Iavernaro, F.; Pace, B., s-stage trapezoidal methods for the conservation of Hamiltonian functions of polynomial type, AIP Conf. Proc., 936, 603-606, (2007) · Zbl 1152.65345
[55] Iavernaro, F.; Pace, B., Conservative block-boundary value methods for the solution of polynomial Hamiltonian systems, AIP Conf. Proc., 1048, 888-891, (2008) · Zbl 1167.65461
[56] Iavernaro, F.; Trigiante, D., High-order symmetric schemes for the energy conservation of polynomial Hamiltonian problems, J. Numer. Anal. Ind. Appl. Math., 4, 1-2, 87-101, (2009) · Zbl 1191.65169
[57] Islas, A. L.; Schober, C. M., On the preservation of phase space structure under multisymplectic discretization, J. Comput. Phys., 197, 2, 585-609, (2004) · Zbl 1064.65148
[58] Islas, A. L.; Schober, C. M., Backward error analysis for multisymplectic discretizations of Hamiltonian pdes, Math. Comput. Simul., 69, 290-303, (2005) · Zbl 1073.65143
[59] Islas, A. L.; Schober, C. M., Conservation properties of multisymplectic integrators, Future Generation Comput. Syst., 22, 412-422, (2006)
[60] Kurganov, A.; Rauch, J., The order of accuracy of quadrature formulae for periodic functions, (Bove, A.; etal., (2009), Birkhäuser Boston)
[61] McLachlan, R. I.; Quispel, G. R.W., Discrete gradient methods have an energy conservation law, Discrete Contin. Dyn. Syst., 34, 3, 1099-1104, (2014) · Zbl 1282.65112
[62] McLachlan, R. I.; Quispel, G. R.W.; Robidoux, N., Geometric integration using discrete gradient, Philos. Trans. R. Soc. Lond. A, 357, 1021-1045, (1999) · Zbl 0933.65143
[63] Laburta, M. P.; Montijano, J. I.; Rández, L.; Calvo, M., Numerical methods for non conservative perturbations of conservative problems, Comput. Phys. Commun., 187, 72-82, (2015) · Zbl 1348.70004
[64] Leimkulher, B.; Reich, S., Simulating Hamiltonian Dynamics, (2004), Cambridge University Press
[65] Li, C. W.; Qin, M. Z., A symplectic difference scheme for the infinite-dimensional Hamilton system, J. Comput. Math., 6, 164-174, (1988) · Zbl 0669.70019
[66] Li, S.; Vu-Quoc, L., Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation, SIAM J. Numer. Anal., 32, 1839-1875, (1995) · Zbl 0847.65062
[67] Lu, X.; Schmid, R., A symplectic algorithm for wave equations, Math. Comput. Simul., 43, 29-38, (1997) · Zbl 0876.65065
[68] Marsden, J. E.; Patrick, G. P.; Shkoller, S., Multi-symplectic geometry, variational integrators, and nonlinear pdes, Commun. Math. Phys., 199, 351-395, (1999)
[69] Matsuo, T., New conservative schemes with discrete variational derivatives for nonlinear wave equations, J. Comput. Appl. Math., 203, 32-56, (2007) · Zbl 1120.65096
[70] Matsuo, T.; Sugihara, M.; Furihata, D.; Mori, M., Spatially accurate dissipative or conservative finite difference schemes derived by the discrete variational method, Jpn. J. Ind. Appl. Math., 19, 3, 311-330, (2002) · Zbl 1014.65083
[71] Mattsson, K.; Nordström, J., Summation by parts operators for finite difference approximations of second derivatives, J. Comp. Phys., 199, 503-540, (2004) · Zbl 1071.65025
[72] Moore, B.; Reich, S., Backward error analysis for multi-symplectic integration methods, Numer. Math., 95, 625-652, (2003) · Zbl 1033.65113
[73] Oliver, M.; West, M.; Wulff, C., Approximate momentum conservation for spatial semidiscretization of semilinear wave equations, Numer. Math., 97, 493-535, (2004) · Zbl 1060.65106
[74] Olsson, P., Summation by parts, projections, and stability: I, Math. Comp., 64, 1035-1065, (1995) · Zbl 0828.65111
[75] Olsson, P., Summation by parts, projections, and stability: I, Math. Comp., 64, 1473-1493, (1995) · Zbl 0848.65064
[76] Qin, M.-Z.; Zhang, M.-Q., Multi-stage symplectic schemes of two kinds of Hamiltonian systems for wave equations, Comput. Math. Appl., 19, 10, 51-62, (1990) · Zbl 0695.65072
[77] Quispel, G. R.W.; McLaren, D. I., A new class of energy-preserving numerical integration methods, J. Phys. A, 41, 045206, 7, (2008) · Zbl 1132.65065
[78] Sanz-Serna, J. M.; Calvo, M. P., Numerical Hamiltonian Problems, (1994), Chapman & Hall
[79] Strand, B., Summation by parts for finite difference approximations for d/dx, J. Comp. Phys., 110, 47-67, (1994) · Zbl 0792.65011
[80] Strauss, W.; Vázquez, L., Numerical solution of a nonlinear Klein-Gordon equation, J. Comput. Phys., 28, 271-278, (1978) · Zbl 0387.65076
[81] Wang, J., A note on multisymplectic Fourier pseudospectral discretization for the nonlinear Schrödinger equation, Appl. Math. Comput., 191, 31-41, (2007) · Zbl 1193.37108
[82] Wineberg, S. B.; Grath, J. F.M.; Gabl, E. F.; Scott, L. R.; Southwell, C. E., Implicit spectral methods for wave propagation problems, J. Comp. Phys., 97, 311-336, (1991) · Zbl 0746.65075
[83] Wlodarczyk, T. H., Stability and Preservation Properties of Multisymplectic Integrators, (2007), Department of Mathematics in the College of Sciences at the University of Central Florida Orlando, Florida, (Ph.D. thesis)
[84] https://www.dm.uniba.it/ testset/testsetivpsolvers/.
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