The Backward Euler Fully Discrete Finite Volume Method for the Problem of Purely Longitudinal Motion of a Homogeneous Bar

Abstract

We present a linear backward Euler fully discrete finite volume method for the initial-boundary-value problem of purely longitudinal motion of a homogeneous bar and an give optimal order error estimates in $L^2$ and $H^1$ norms. Furthermore, we obtain the superconvergence error estimate of the generalized projection of the solution $u$ in $H^1$ norm. Numerical experiment illustrates the convergence and stability of this scheme.