An Application of JFNK Method to Solving the 1D Nonlinear Advection Equation in Fully Implicit Scheme
JFNK方法在求解全隐式一维非线性平流方程中的应用

Over the last 10 years it had major advantages in super computing that the higher and higher resolution was adopted in atmospheric numerical models for various scales.And this in turn provides possibility and necessity for using an implicit scheme.Zeng and Ji proposed a scheme with energy conservation in 1981.However,the convergence of iteration involved in the fully implicit scheme remains a key issue to solve.JFNK(Jacobian-free Newton-Krylov) method,a highly efficient method for solving implicit nonlinear equation,has been successfully used in many fields of computational physics.JFNK method is a nested iteration method for solving the nonlinear partial differential equations(PDEs),which is synergistic combination of Newton-type method and Krylov subspace method,without forming and storing the elements of the true Jacobian matrix.Some studies have shown that the fully implicit method solved by JFNK is more efficient and accurate than the semi-implicit method for a given level of accuracy and for a given CPU time.This paper,based on the 1D nonlinear advection equation(namely the inviscid Burgers equation),a number of comparative simulations,with the 2-order fully implicit(IM) schemes,the 2-order explicit(EX) schemes and 2-order semi-implicit(SI) schemes,have been carried out to investigate the necessity of adopting fully implicit schemes and the effect of JFNK method.The results show that the implicit scheme has better performance in comparison with the other two schemes in terms of computational stability and accuracy,especially for the simulation of flow system with strong current or large gradient of variables.These can be summarized as follows.(1) When the time step is smaller(say time step is 2-9),all the three schemes are stable;but the accuracy of IM scheme is higher than it of the others.And with the increase of time step,the IM scheme remains stable,while the other two become unstable up to time step is 2-3 when basic flow is 0.0,time step is 2-4 when basic flow is 1.0,and time step is 2-5 when basic flow is 2.0.(2) With the increase of basic flow,the advantage of IM scheme over the others becomes even more remarkable with respect to the accuracy and stability.(3) In the region of sharp gradient of variable,the IM scheme still shows clearly smooth and accurate simulation,while for the SI and EX schemes false perturbations occur there and disperse both upstream and downstream gradually.The successful application of JFNK methods on the inviscid burgers equation shows the prospect for the application of the method to more complicated nonlinear equation in future.