Analytical solutions for nonlinear long wave propagation

A two-parameter homotopy method is proposed for general nonlinear problems and demonstrated through continuous solutions of long waves with dispersive terms and discontinuous solutions without dispersive terms. This analytical method is based on a new zero-order deformation equation in topology that includes an auxiliary operator for mapping of an initial approximation to the exact solution and two auxiliary parameters to ensure fast convergence of the series solution. Residual error analysis through algebraic and transcendental equations shows that the proposed zero-order deformation equation substantially improves the convergence region and rate of the series solution and allows greater freedom in the selection of auxiliary operators than the traditional homotopy method.We derive a general series solution for nonlinear differential equations based on the two-parameter homotopy method. Implementation of the general solution is demonstrated with the periodic dispersive long-wave problem governed by the Korteweg de Vries equation and the propagation of high-frequency waves in a relaxing medium given by the Vakhnenko equation. Comparison of the present and exact solutions confirms the effectiveness and validity of the proposed approach. The use of two auxiliary parameters substantially improves the convergence region and rate and provides series solutions to highly nonlinear equations with fewer terms.The Riemann solver is the fundamental building block in the Godunov-type formulation of many nonlinear long-wave problems involving discontinuities. While existing solvers are obtained either iteratively or through approximations of the Riemann problem, this study reports an explicit analytical solution to the nonlinear algebraic equations of the exact Riemann problem through the homotopy method. A sensitivity analysis shows fast convergence of the series solution and the first three terms provide highly accurate results. The proposed Riemann solver is implemented in an existing finite-volume model with a Godunov-type scheme. The model correctly describes the formation of shocks and rarefaction fans for both one and two-dimensional dam-break problems, thereby verifying the proposed Riemann solver for general implementation.This study has demonstrated the power of the homotopy method in obtaining analytical solutions for highly nonlinear problems and the feasibility of implementing analytical solvers in numerical solutions for general applications.