A Framework for the Numerical Solution of Klein–Gordon Eequations Using Three New Collocation Methods

Subject Areas : International Journal of Mathematical Modelling & Computations

Bahareh Sadeghi ¹

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Received: 2024-05-15 Accepted : 2024-08-15 Published : 2024-08-22

Keywords: Klein−Gordon equations, Legendre interpolation, Legendre−Gauss−Radau collocation, Domain decomposition, Quasilinearization.,

Abstract :

The Klein−Gordon equation is a relativistic version of the Schr¨odinger equation and has a large range of applications in contemporary physics. In this paper, we present a unified framework for the numerical solution of nonlinear Klein−Gordon equations using three new collocation schemes. The solutions are approximated by the two−dimensional Legendre−Gauss−Radau interpolation directly. Moreover, using the properties of Jacobi polynomials, the partial derivatives of the solutions are expressed in terms of Jacobi polynomials that makes the approximations numerically more stable. We first derive a single−domain collocation method (SDC) that is suited for problems in small and moderate domains. Next, a multi−domain collocation method (MDC) is presented for large domains. The proposed MDC method solves the problem step by step in subdomains whilst is of the BN−stability. We then construct a single−domain iterative collocation method (SDIC). The proposed SDIC method is based on the quasilinearization (QL) technique and is suited for highly nonlinear problems. The key properties of the SDC, MDC and SDIC schemes are explained and the convergence of the QL approach applied to PDEs with the differential operator of the wave equation is assessed. Numerical examples are included to assess the accuracy and features of each collocation scheme.

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