Comparison of exponential integrators and traditional time integration schemes for the shallow water equations

Brachet, Matthieu; Debreu, Laurent; Eldred, Christopher

doi:10.1016/j.apnum.2022.05.006

Cited by 7 publications

(5 citation statements)

References 37 publications

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“…This is one of the inherent advantages of our method that the stability of the system of ODEs is driven only by the underlying constant matrix and is not influenced by an additional effect from any time discretisation in the model. For a detailed stability analysis of solving such a special system of equations through a method of exponential integrators, the reader can refer to Brachet et al (2020) and Buvoli and Minion (2022).…”

Section: Study Of Convergence and Stabilitymentioning

confidence: 99%

Method of Lines for Valuation and Sensitivities of Bermudan Options

2022

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In this paper, we present a computationally efficient technique based on the Method of Lines for the approximation of the Bermudan option values via the associated partial differential equations. The method of lines converts the Black Scholes partial differential equation to a system of ordinary differential equations. The solution of the system of ordinary differential equations so obtained only requires spatial discretization and avoids discretization in time. Additionally, the exact solution of the ordinary differential equations can be obtained efficiently using the exponential matrix operation, making the method computationally attractive and straightforward to implement. An essential advantage of the proposed approach is that the associated Greeks can be computed with minimal additional computations. We illustrate, through numerical experiments, the efficacy of the proposed method in pricing and computation of the sensitivities for a European call, cash-or-nothing, powered option, and Bermudan put option.

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Section: Study Of Convergence and Stabilitymentioning

confidence: 99%

Method of Lines for Valuation and Sensitivities of Bermudan Options

2022

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“…A particular class of methods, known as time exponential integrators, have been shown to outperform classical schemes in several partial differential equation (PDE) models from several applied areas in terms of accuracy and computational performance (e.g., [7,10,19,24,26,31]). Such methods are known to accurately describe the solution of linear waves and, therefore, seem like natural candidates for seismic wave propagation problems.…”

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confidence: 99%

An explicit exponential time integrator based on Faber polynomials and its application to seismic wave modelling

V.¹,

Peixoto²,

Schreiber³

2022

Preprint

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Exponential time integrators have been applied successfully in several physics-related differential equations. However, their application in hyperbolic systems with absorbing boundaries, like the ones arising in seismic imaging, still lacks theoretical and experimental investigations.The present work conducts an in-depth study of exponential integration using Faber polynomials, consisting of a generalization of a popular exponential method that uses Chebyshev polynomials. This allows solving non-symmetric operators that emerge from classic seismic wave propagation problems with absorbing boundaries. Theoretical as well as numerical results are presented for Faber approximations. One of the theoretical contributions is the proposal of a sharp bound for the approximation error of the exponential of a normal matrix. We also show the practical importance of determining an optimal ellipse encompassing the full spectrum of the discrete operator, in order to ensure and enhance convergence of the Faber exponential series. Furthermore, based on estimates of the spectrum of the discrete operator of the wave equations with a widely used absorbing boundary method, we numerically investigate stability, dispersion, convergence and computational efficiency of the Faber exponential scheme.Overall, we conclude that the method is suitable for seismic wave problems and can provide accurate results with large time step sizes, with computational efficiency increasing with the increase of the approximation degree.

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“…A particular class of methods, known as exponential integrators, have been shown to outperform classical schemes in several partial differential equations (PDEs) models from several applied areas in terms of accuracy and computational performance (e.g., [9,14,28,34,36,44]). For the specific case of linear differential equations, there have been developed specialized exponential integrators.…”

Section: Chaptermentioning

confidence: 99%

“…To approximate the solutions using Faber polynomials, we define a spatial step size ∆x = 0.0025 for the 1D examples and ∆x = 0.02 for the 2D examples and use a finite difference scheme with 4th and 8th spatial order. For comparison purposes, we use a nine-stage seventh-order temporal Runge-Kutta scheme RK (9,7) recommended for hyperbolic problems (see Calvo et al [11])) with a small time-step size (∆t = ∆x/(8c max )), where c max is the maximum velocity. The formulas of the RK(9-7) algorithm are presented in Appendix A.6.…”

Section: Convergence and Efficiencymentioning

confidence: 99%

“…The formulas of the RK(9-7) algorithm are presented in Appendix A.6. The equation formulation and spatial discretization size and order used in the RK (9,7) are the same as the one adopted in the Faber approximations, so the spatial operator is exactly the same as the one used in the Faber approximation scheme. Therefore, only temporal effects are visible in the numerical errors shown.…”

Section: Convergence and Efficiencymentioning

confidence: 99%

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On explicit exponential integrators in the solution of elastic wave propagation equations

Ravelo

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Comparison of exponential integrators and traditional time integration schemes for the shallow water equations

Cited by 7 publications

References 37 publications

Method of Lines for Valuation and Sensitivities of Bermudan Options

Method of Lines for Valuation and Sensitivities of Bermudan Options

An explicit exponential time integrator based on Faber polynomials and its application to seismic wave modelling

On explicit exponential integrators in the solution of elastic wave propagation equations

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