A scheme is outlined for solving hyperbolic conservation laws by finite element method of piecewise linear interpolations. It is different from the discontinuous finite element on the boundaries of neighboring cells to solve Riemann problems that the scheme is designed to solve hyperbolic conservation laws based on the Hamilton Jacobi equations. Under the CFL condition, the scheme is proved that it satisfies the maximal principle and is a TVD scheme. Numerical examples are given and discussed.

In multicomponent flow, different component has diffe rent equation of states. This makes the flux discontinuous and no Jacobian matrix exists. In schemes of high resolution for Euler equations, the Jacobian matrix and its eigenvalues as while as its eigenvectors are needed, that is, the flux should be continuously differentiable. So the whole systems of conservation laws should be rearranged. For γ gas, γ is regarded as a new unknown and a new con servation equation is added, thus the flux of the new system become continuously differentiable, the obstacles in the way to high resolution schemes are removed . Since true flows only obey three conservation laws, though the additional cons ervation law does not influence the exact solutions of original differential equ ations, it does influence the numerical solutions of difference equations. This is obvious in numerical experiments. The presented method eliminates this influe nce as much as possible. All schemes of single component flow can be directly us ed. Numerical experiments for one dimensional shock tube problem of multicompone nt flow demonstrate that so designed schemes have the same effect as those of on e component flow.