Study On The Exact Solution Method Of Riemann Problem For Shallow Water Equation

The discontinuous problem of the Shallow Water Equations(SWE)is called Riemann Problem(RP).Riemann problem includes homogeneous Riemann problem and non-homogeneous Riemann problem.The Riemann Problem with the slope source term is called Step Riemann Problem(SRP).The homogeneous Riemann problem has been fully studied and its solution methods include approximate Riemann solver and exact Riemann solver.The former includes Roe,HLL and ASUM,etc.,and the second can be done by simple Newton iteration method,all of which have achieved good results.However,the SRP is unresolved.This paper tries to solve this problem by using the iterative method.The main work is as follows:(1)SRP is to derive two new states on the basis of the initial conditions.There is a mathematical relationship between these two new states and the initial conditions on both sides.If the relation is presented as shock wave,the RH condition is satisfied between the physical quantity of the new derivative state and the physical quantity of the initial condition.If the relation is presented in the form of rarefaction waves,the generalized Riemann condition is satisfied between the physical quantity of the new derived state and the physical quantity of the initial condition.The combination of these two conditions forms four Riemann solutions.Unlike the homogeneous Riemann problem,there is a relation between the physical quantities in the two new derivative states of the stepped Riemann problem,which satisfies the RH condition,and will be presented in the form of static shock waves.In addition,a method to simplify the equations is proposed to speed up the solution of the equations.(2)The problems that appear when the Newton method is used to solve the above equations are studied and discussed.These problems reflect the limitations of the Newton method in solving the Riemann problem of shallow water equation ladder.Newton iteration method converges to the wrong solution;In the process of the iteration of Newton method,the jacobian matrix appears singular or nearly singular,and the iteration is difficult to continue.In view of the limitations of Newton method,two improved algorithms are sought in other literatures:the midpoint quadrature method and the continuous modified Newton method to avoid the disadvantages of Newton method.The midpoint quadrature method is used to optimize the initial value of iteration to improve Newton method.The continuous modified Newton law can be improved by optimizing the iteration length.Examples of the two algorithms are given under different Riemannian solutions.(3)The paper discusses the gauss-newton optimization algorithm,the damped Newton algorithm,or LM algorithm,and studies the numerical characteristics of the algorithm.The selection of damper factors and corresponding matching strategies of LM method are introduced,and the adaptive LM algorithm is constructed by adjusting the adaptive factors in combination with the trust region method.The iterative process of adaptive LM algorithm is explained in detail.It can be seen from the numerical process that,by adjusting the damping factor,the adaptive LM method has the ability to avoid the Jacobi matrix to approximate the singular region,so as to calculate the difficult nonlinear equations.The algorithm is verified under the condition of poor initial value of iteration.