By solving the inverse generalized Riemann problem for the quasilinear hyperbolic system of conservation laws, the exact shock reconstruction is realized in both local and global sense.

Structured computational grids are the basis for highly efficient numerical approximations of wave propagation. When there are discontinuous material coefficients the accuracy is typically reduced and there may also be stability problems. In a sequence of recent papers Gustafsson et al. proved stability of the Yee scheme and a higher order difference approximation based on a similar staggered structure, for the wave equation with general coefficients. In this paper, the Yee discretization is improved from first to second order by modifying the material coefficients close to the material interface. This is proven in the $L^2$ norm. The modified higher order discretization yields a second order error component originating from the discontinuities, and a fourth order error from the smooth regions. The efficiency of each original method is retained since there is no special structure in the difference stencil at the interface. The main focus of this paper is on one spatial dimension, with the derivation of a second order algorithm for a two dimensional example given in the last section.

In this paper, we shall establish the well-posedness of a mathematical model for a special class of electrochemical power device – lithium-ion battery. The underlying partial differential equations in the model involve a (mix and fully) coupled system of quasi-linear elliptic and parabolic equations. By exploring some special structure, we are able to adopt the well-known Nash-Moser- DeGiorgi boot strap to establish suitable a priori supremum estimates for the electric potentials. Using the supremum estimates, we apply the Leray-Schauder theory to establish the existence and uniqueness of a subsystem of elliptic equations that describe the electric potentials in the model. We then employ a Schauder fix point theorem to obtain the local (in time) existence for the whole model. We also consider the global existence of a modified 1-d governing system under additional assumptions. In particular, we are able to derive uniform a priori estimates depending only on the existence time $T$, including the supremum estimates for electric potentials and growth and decay estimates for the concentration $c$. Using the uniform estimates, we prove that the modified system has a solution for all time $t>0$.

Fast sweeping methods utilize the Gauss-Seidel iterations and alternating sweeping strategy to achieve the fast convergence for computations of static Hamilton-Jacobi equations. They take advantage of the properties of hyperbolic PDEs and try to cover a family of characteristics of the corresponding Hamilton-Jacobi equation in a certain direction simultaneously in each sweeping order. The time-marching approach to steady state calculation is much slower than the fast sweeping methods due to the CFL condition constraint. But this kind of fixed-point iterations as time- marching methods have explicit form and do not involve inverse operation of nonlinear Hamiltonian. So it can solve general Hamilton-Jacobi equations using any monotone numerical Hamiltonian and high order approximations easily. In this paper, we adopt the Gauss-Seidel idea and alternating sweeping strategy to the time-marching type fixed-point iterations to solve the static Hamilton-Jacobi equations. Extensive numerical examples verify at least a $2\sim5$ times acceleration of convergence even on relatively coarse grids. The acceleration is even more when the grid is further refined. Moreover the Gauss-Seidel philosophy and alternating sweeping strategy improves the stability, i.e., a larger CFL number can be used. Also the computational cost is exactly the same as the time-marching scheme at each time step.