  In this work a new finite element based Method of Relaxed Streamline
Upwinding is proposed to solve hyperbolic conservation laws. Formulation of the
proposed scheme is based on relaxation system which replaces hyperbolic
conservation laws by semi-linear system with stiff source term also called as
relaxation term. The advantage of the semi-linear system is that the
nonlinearity in the convection term is pushed towards the source term on right
hand side which can be handled with ease. Six symmetric discrete velocity
models are introduced in two dimensions which symmetrically spread foot of the
characteristics in all four quadrants thereby taking information symmetrically
from all directions. Proposed scheme gives exact diffusion vectors which are
very simple. Moreover, the formulation is easily extendable from scalar to
vector conservation laws. Various test cases are solved for Burgers equation
(with convex and non-convex flux functions), Euler equations and shallow water
equations in one and two dimensions which demonstrate the robustness and
accuracy of the proposed scheme. New test cases are proposed for Burgers
equation, Euler and shallow water equations. Exact solution is given for
two-dimensional Burgers test case which involves normal discontinuity and
series of oblique discontinuities. Error analysis of the proposed scheme shows
optimal convergence rate. Moreover, spectral stability analysis gives implicit
expression of critical time step.
