JFNK Method And Its Application In The Fully Implicit Nonlinear Inviscid Burgers Equation

JFNK (Jacobian-Free Newton-Krylov) methods are the nested iteration methods for solving the nonlinear partial differential equations (PDEs),which are synergistic combinations of Newton-type methods for super-linearly convergent solution of nonlinear equations and Krylov subspace methods for solving the Newton correction equations, without forming and storing the elements of the true Jacobian matrix. This paper briefly introduces the JFNK method and employs it to solve the fully implicit inviscid Burgers equation. The numerical solutions of the implicit equation is compared to that of the explicit and semi-Lagrangian equations, and the advantages of implicit equation in simulating the great gradient phenomena are analyzed as the emphasis. the results show that:(1) The numerical solution of the fully implicit inviscid Burgers equation solved by JFNK method is convergent, which indicates that it is feasible using JFNK method solve the fully implicit nonlinear PDEs because that the method gets over the convergence problem of numerical solution of the equations.(2) The tine steps of the implicit equation are unlimited by the spatial steps because the fully implicit method is unconditionally stable, Compared to the analytical solutions, its numerical solutions well consistent to the analytical solutions. specially the numerical solutions are smooth as same as the analytical solutions, while the solutions of the explicit and semi-Lagrangian equations are not.(3) Analysis of the impact of â–³x,â–³t on the computation quantity of JFNK method suggests that the iteration number, the computation and store demand of everyiteration decrease following the â–³x 's decreasing, while the iteration speed increases following the â–³t's decreasing.(4) Employed the same â–³x and small enough â–³t, the accuracy of explicit method and the accuracy of semi-Lagrangian method have not apparently different in whole area, while explicit method better than them, and at GGA the advantage of implicit method is more obvious. And, employed same longer â–³t, the accuracy of implicit method and the accuracy of semi-Lagrangian method have not apparently different in whole area, while explicit method better than them, but at GGA the former is better than the latter too.(5) The mean root square error of implicit method increases following the â–³x's decreasing while that of explicit and semi-Lagrangian method decreases following the Ax 's decreasing. The error of implicit method simply decreases following the Ax 's decreasing while that of explicit and semi-Lagrangian method is complex following the Ax 's decreasing.