Search

In this paper, a reliable and effective local discontinuous Galerkin (LDG) scheme for numerically solving the classical Rosenau–Hyman equation with non-periodic boundary conditions has been proposed. This study employs the third-order nonlinearly stable total variation diminishing Runge–Kutta method and the LDG method, respectively, to discretize the temporal and spatial derivatives. Finally, numerical simulations are performed on various test problems and compared with the exact results as well as results produced by a few other numerical methods, to analyze the reliability and efficiency of the proposed method. The results generated, which validate the expected order of accuracy, are presented through multiple tables. In addition, several graphical representations of the problem are presented to depict the behavior of the solution. [ABSTRACT FROM AUTHOR]

The development of an effective and efficient local discontinuous Galerkin algorithm for the solution of the coupled (1 + 1) -dimensional Broer–Kaup system is the primary focus of this research. In this algorithm, spatial discretization is attained using the local discontinuous Galerkin method, while temporal discretization is handled using the explicit total variation diminishing higher-order Runge-Kutta method. However, the ( G ′ G ) -expansion method is also implemented to produce the exact solutions for the coupled (1 + 1) -dimensional Broer–Kaup system. The obtained solutions are the traveling wave solutions. The efficiency and reliability of the proposed method are analyzed by comparing the generated numerical simulations to various traveling wave solutions using several tables and figures. The exact solutions and the numerical simulations of the coupled (1 + 1) -dimensional Broer–Kaup system are shown to correspond exceptionally well. [ABSTRACT FROM AUTHOR]

This paper investigates numerical solutions for the unperturbed and general perturbed Newell–Whitehead–Segel-type equations with the help of the local discontinuous Galerkin method. The stability analysis and error estimations of the proposed local discontinuous Galerkin algorithm are extensively examined. First, the spatial variables are discretized to provide a semidiscrete method of lines scheme. This generates an ordinary differential equation system in the temporal variable, which is subsequently solved using the total variation diminishing Runge–Kutta method of higher order. The generated numerical results are compared to the exact results and a few other existing numerical methods via various tables and figures to illustrate the efficiency and accuracy of the proposed method. The numerical results show that the proposed method is an effective numerical scheme for solving the Newell–Whitehead–Segel equation since the solutions obtained using the local discontinuous Galerkin method are highly close to the exact solutions with significantly less error. [ABSTRACT FROM AUTHOR]

In this paper, the local discontinuous Galerkin method is used to analyze numerical solutions for nonlinear Allen–Cahn equations with nonperiodic boundary conditions. To begin with, the spatial variables are discretized to generate a semidiscrete method of lines scheme. This yields an ordinary differential equation system in the temporal variable, which is then solved using the higher-order total variation diminishing Runge–Kutta method. A comparison of the generated numerical results to the exact results for various test problems using different tables and figures provides insight into the effectiveness and accuracy of the proposed method. The numerical results confirm that the proposed method is an effective numerical scheme for solving the Allen–Cahn equation since the obtained solutions are extremely close to the exact solutions while exhibiting substantially less error. [ABSTRACT FROM AUTHOR]