Analysis of a Fourier-Galerkin numerical scheme for a 1D Benney-Luke-Paumond equation

We study convergence of the semidiscrete and fully discrete formulations of a Fourier- Galerkin numerical scheme to approximate solutions of a nonlinear Benney-Luke-Paumond equation that models long water waves with small amplitude propagating over a shallow channel with at bottom. The accuracy of the numerical solver is checked using some exact solitary wave solutions. In order to apply the Fourier-spectral scheme in a non periodic setting, we approximate the initial value problem with x ∈ R by the corresponding periodic Cauchy problem for x ∈ [0, L], with a large spatial period L.

We study convergence of the semidiscrete and fully discrete formulations of a Fourier-Galerkin numerical scheme to approximate solutions of a nonlinear Benney-Luke-Paumond equation that models long water waves with small amplitude propagating over a shallow channel with at bottom. The accuracy of the numerical solver is checked using some exact solitary wave solutions. In order to apply the Fourier-spectral scheme in a non periodic setting, we approximate the initial value problem with x ∈ R by the corresponding periodic Cauchy problem for x ∈ [0, L], with a large spatial period L.

DOI: https://doi.org/10.15446/recolma.v49n2.60440

Análisis de un esquema numérico Fourier-Galerkin para una ecuación unidimensional Benney-Luke-Paumond

Juan Carlos Muñoz Grajales¹

¹ Universidad del Valle, Cali, Colombia
e-mail: jcarlmz@yahoo.com

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Abstract

We study convergence of the semidiscrete and fully discrete formulations of a Fourier-Galerkin numerical scheme to approximate solutions of a nonlinear Benney-Luke-Paumond equation that models long water waves with small amplitude propagating over a shallow channel with flat bottom. The accuracy of the numerical solver is checked using some exact solitary wave solutions. In order to apply the Fourier-spectral scheme in a non periodic setting, we approximate the initial value problem with x ∈ ℝ by the corresponding periodic Cauchy problem for x ∈ [0,L], with a large spatial period L.

Key words and phrases. Solitary waves, water waves, spectral methods.

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2010 Mathematics Subject Classification. 35Q35, 35B35, 76B25, 65N35.

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Resumen

Estudiamos la convergencia de las formulaciones semidiscreta y completamente discreta de un método espectral Fourier-Galerkin para aproximar las soluciones de una ecuación no lineal Benney-Luke-Paumond que modela ondas largas con pequeña amplitud que se propagan sobre un canal raso con fondo plano. La precisión del método numérico se verifica usando algunas soluciones de onda solitaria exactas. A fin de aplicar el esquema Fourier-espectral en un contexto no periódio, aproximamos el problema de valor inicial con x ∈ ℝ por el correspondiente problema de Cauchy periódico para x ∈ [0,L], con un periodo espacial L grande.

Palabras y frases clave: Ondas solitarias, ondas acuáticas, métodos espectrales.

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References

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(Recibido en junio de 2015. Aceptado en octubre de 2015)