Stability analysis for a fully discrete spectral scheme for Boussinesq systems

Xavier, Juliana Castanon; Rincon, Mauro A.; Vigo, Daniel G. Alfaro; Amundsen, David E.

doi:10.1080/00036811.2017.1281405

Cited by 11 publications

(9 citation statements)

References 19 publications

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“…• Case (vii): b > 0, d = 0, a < 0, c = 0. As mentioned in the Introduction, the systems of cases (i) (BBM-BBM) and (v) ('Bona-Smith') have been discretized by a collocation spectral method in space an analyzed by Xavier et al, [68], in the case of surface waves. The error estimates obtained in [68] are similar to those that we obtain below for the spectral Galerkin method in these cases but we include the proofs as our techniques are somewhat different.…”

Section: Error Estimates For a Spectral Semidiscretization Of The Per...mentioning

confidence: 99%

“…The papers [10] and [9] also contain error estimates of temporal discretizations of the semidiscrete problems effected with high-order accurate, explicit Runge-Kutta (RK) time-stepping schemes. In [68] Xavier et al analyze spectral methods of collocation type, coupled with the explicit, 'classical', fourth-order accurate RK scheme for time-stepping for the surface-wave Boussinesq systems corresponding to the classes of the cases (i) and (v), see section 2 in the sequel.…”

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confidence: 99%

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Notes on numerical analysis and solitary wave solutions of Boussinesq/Boussinesq systems for internal waves

Dougalis,

Duran,

Saridaki

2020

Preprint

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In this paper a three-parameter family of Boussinesq systems is studied. The systems have been proposed as models of the propagation of long internal waves along the interface of a two-layer system of fluids with rigid-lid condition for the upper layer and under a Boussinesq regime for the flow in both layers. The contents of the paper are as follows. We first present some theoretical properties of well-posedness, conservation laws and Hamiltonian structure of the systems, using the results for analogous models for surface wave propagation. Then the corresponding periodic initial-value problem is discretized in space by the spectral Fourier Galerkin method and for each well posed system, error estimates for the semidiscrete approximation are proved. The rest of the paper is concerned with the study of existence and the numerical simulation of some issues of the dynamics of solitary-wave solutions. Standard theories are used to derive several results of existence of classical and generalized solitary waves, depending on the parameters of the models. A numerical procedure based on a Fourier collocation approximation for the ode system of the solitary wave profiles with periodic boundary conditions, and on the iterative solution of the resulting fixed-point systems with the Petviashvili scheme combined with vector extrapolation techniques, is used to generate numerically approximations of solitary waves. These are taken as initial conditions in a computational study of the dynamics of the solitary waves, both classical and generalized. To this end, the spectral semidiscretizations of the periodic initial-value problem for the systems are numerically integrated by a fourth-order Runge-Kutta-composition method based on the implicit midpoint rule. The fully discrete scheme is then used to approximate the evolution of small and large perturbations of computed solitary wave profiles, and to study computationally the collisions of solitary waves as well as the resolution of initial data into trains of solitary waves.

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Section: Error Estimates For a Spectral Semidiscretization Of The Per...mentioning

confidence: 99%

mentioning

confidence: 99%

Notes on numerical analysis and solitary wave solutions of Boussinesq/Boussinesq systems for internal waves

Dougalis,

Duran,

Saridaki

2020

Preprint

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“…Several methods have been devised for the numerical solution of relevant initial-boundary value problems for the system (1). These include, for example, spectral [35,28], finite volume/difference [22,12] and Galerkin finite element methods [7,6], where they have been analysed, tested and studied in depth. Some of these numerical methods appear to have good conservation and accuracy properties but none of them has been designed to conserve the energy functional (3).…”

Section: Introductionmentioning

confidence: 99%

A Conservative Fully Discrete Numerical Method for the Regularized Shallow Water Wave Equations

Mitsotakis

Ranocha²,

Ketcheson

et al. 2021

SIAM J. Sci. Comput.

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The paper proposes a new, conservative fully-discrete scheme for the numerical solution of the regularised shallow water Boussinesq system of equations in the cases of periodic and reflective boundary conditions. The particular system is one of a class of equations derived recently and can be used in practical simulations to describe the propagation of weakly nonlinear and weakly dispersive long water waves, such as tsunamis. Studies of small-amplitude long waves usually require long-time simulations in order to investigate scenarios such as the overtaking collision of two solitary waves or the propagation of transoceanic tsunamis. For long-time simulations of non-dissipative waves such as solitary waves, the preservation of the total energy by the numerical method can be crucial in the quality of the approximation. The new conservative fullydiscrete method consists of a Galerkin finite element method for spatial semidiscretisation and an explicit relaxation Runge-Kutta scheme for integration in time. The Galerkin method is expressed and implemented in the framework of mixed finite element methods. The paper provides an extended experimental study of the accuracy and convergence properties of the new numerical method. The experiments reveal a new convergence pattern compared to standard Galerkin methods. E(t; η, u) := 1 2 I gη 2 + (D + η)u 2 dx, (3)

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“…Concerning the numerical approximation of systems of Boussinesq type for surface and internal waves, error estimates of Galerkin-Finite Element semidiscrete schemes and for associated fully discrete schemes with high-order, explicit Runge-Kutta (RK) methods for some initial-boundary-value problems for several surfacewave Boussinesq systems can be found in [4,5,13]. Spectral methods of collocation type with the classical explicit 4th-order RK time integrator to discretize some Boussinesq systems in the surface wave case are analyzed in [27], while the L 2 convergence of semidiscrete approximations of the Boussinesq/Boussinesq systems modelling internal wave propagation with the spectral Fourier-Galerkin method is established in [11]. A full discretization of the resulting semidiscrete systems with a fourth-order RK Composition method is studied computationally in the previous paper and its extended version [12].…”

Section: Introductionmentioning

confidence: 99%

Notes on the Boussinesq-Full dispersion systems for internal waves: Numerical solution and solitary waves

Dougalis¹,

Durán²,

Saridaki³

2021

Preprint

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In this paper we study some theoretical and numerical issues of the Boussinesq/Full dispersion system. This is a a three-parameter system of pde's that models the propagation of internal waves along the interface of two-fluid layers with rigid lid condition for the upper layer, and under a Boussinesq regime for the upper layer and a full dispersion regime for the lower layer. We first discretize in space the periodic initial-value problem with a Fourier-Galerkin spectral method and prove error estimates for several ranges of values of the parameters. Solitary waves of the model systems are then studied numerically in several ways. The numerical generation is analyzed by approximating the ode system with periodic boundary conditions for the solitary-wave profiles with a Fourier spectral scheme, implemented in a collocation form, and solving iteratively the corresponding algebraic system in Fourier space with the Petviashvili method accelerated with the minimal polynomial extrapolation technique. Motivated by the numerical results, a new result of existence of solitary waves is proved. In the last part of the paper, the dynamics of these solitary waves is studied computationally, To this end, the semidiscrete systems obtained from the Fourier-Galerkin discretization in space are integrated numerically in time by a Runge-Kutta Composition method of order four. The fully discrete scheme is used to explore numerically the stability of solitary waves, their collisions, and the resolution of other initial conditions into solitary waves.

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Stability analysis for a fully discrete spectral scheme for Boussinesq systems

Cited by 11 publications

References 19 publications

Notes on numerical analysis and solitary wave solutions of Boussinesq/Boussinesq systems for internal waves

Notes on numerical analysis and solitary wave solutions of Boussinesq/Boussinesq systems for internal waves

A Conservative Fully Discrete Numerical Method for the Regularized Shallow Water Wave Equations

Notes on the Boussinesq-Full dispersion systems for internal waves: Numerical solution and solitary waves

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