Some Modifications Of Jacobian-Free Newton-Krylov Methods And Their Applications In 2-D Numerical Simulation Of River Flow

How to speed up computing rate is one of the most important problems in 2-Dnumerical simulation of unsteady river flow. Based on the current discretizingtecheniches, it is helpful for improving computing rate to find efficient solvers ofnonlinear systems arising from the discretization of the governing equations.Jacobian-free Newton–Krylov (JFNK) methods, which have been developed in recentyears, can be viewed as the robust and efficient coupled algorithms to solve the largesparsesystems ofnonlinearequations. Thesemethods aresynergisticcombinations ofinexact Newton's method for super-linearly convergent solution of nonlinearequations and Krylov subspace methods for solving the Newton correction equations.The link between the two methods is the Jacobian-vector product, which may beprobed approximately without forming and storing the elements of the true Jacobianmatrix, through a variety of means. JFNK has potential for 2-D numerical simulationof unsteady river flow. Successful application of the JFNK methods to any givenproblem is dependent on adequate choice of the forcing terms and preconditioning.Here,we present some modifications of JFNK methods. Furthermore, we use JFNKmethods with these modifications for solving the nonlinear systems of equationsarising from the implicit discretization of the governing equations. The maincontributionscanbelistedbelow:1. We provide an overview of the mechanics of JFNK. We analyze thecharacteristic of Jacobian matrix of nonlinear systems arising from the discretizationof the governing equations. When JFNK methods are employed for solve theproblems considered here, analysis of convergence shows that the methods canconvergerapidly.2. We give some remarks on the known strategies for the choice of forcing terms.Based on these remarks, we present a new choice, which is suitable for the nonlinearpreconditioned JFNK and two-dimensional numerical simulation of unsteady riverflow.3. Due to the problems that the Jacobian arising from two-dimensional river modelis numerically singular or ill-conditioned, we present a new modified Newton'smethod, which can solve such problems and has the same order of convergence asNewton's method. We also present a simplified form of this method and a modified SSOR-Newtonmethod,whichmaybecombinedwithpreconditioning.4. Based on the characteristic of Jacobian arising from two-dimensional numericalsimulationofunsteadyriverflow,wecanconcludethatthechoiceofrelaxationfactoris very necessary for SSOR preconditioner. But the known strategies for choice ofrelaxation factor can not be used in our problems. Here, we present a new choice ofrelaxation factor. The numerical results show that SSOR preconditioner with thischoicecanmakeJFNKconvergemorerapidlyandrequirelessmemory.5. Due to less memory requirements, we present an efficient nonlinearpreconditioning technique, and then based on this technique, construct a particularnonlinearSSOR-Newtonpreconditioner,whichrequireslessmemory.6.Basedonthepresent choiceoftheforcingterms and preconditioningtechniques,afamilyofmatrix-freepreconditionedJFNKmethodshasbeendevelopedtosolvethenonlinear systems of equations arising from the fully implicit discretization of thegoverning equations. We test a river model with enlarged lower reach, and thenumerical results demonstrate that the new methods are fast, robust and require lessmemory.Thenumerical resultsshowthat:thepresentchoiceoftheforcingtermscanimprove the convergence and robustness, especially in the case that the nonlinearpreconditioner is used; the presnt approximately optical Linear SSOR preconditionerand Nonlinear SSOR-Newton preconditioner can make JFNK faster and more robust,and require less memory; the former is more efficient than the current LU-SGSpreconditioner,whilethelattercanuselongertimestepthantheformer,thereforeitispreferableforlongtimesimulationofriverflow.