RKMK Geometric Integration Of Non-linear Partial Differential Equations

Many mathematicians, physicists and engineers have been interested in mathematical and mechanical theory and its corresponding computational method on non-linear dynamical system. And they have become the popular research area on the foundation and the applied application in the world now. We study Runge-Kutta/Munthe-Kaas (RKMK) method of non-linear dynamical equation. We compound the computation of exponential matrix with the classical Runge-Kutta method, and present a kind of simple effective RKMK integration method in the Minkowski space. It belongs to Lie group method.In this paper, we first use the RKMK geometric integration method to solve the non-damping Landau-Lifshitz equation and compare its numerical results with the exact solution of the equation. Secondly, we use the RKMK method to solve Landau-Lifshitz equation with the magnetic field and compare its error with that of the classical Runge-Kutta method. After the comparison, we find that the RKMK method can maintain the square conservation characteristic better than the classical Runge-Kutta method. Finally, we use the RKMK method to solve the non-linear Schrodinger equation with variable coefficient.