Estimates away from a discontinuity for dissipative Galerkin methods for hyperbolic equations

Abstract:

We consider the approximate solution of the initial value problem \[ \frac {{\partial u}}{{\partial t}} = \frac {{\partial u}}{{\partial x}},\quad u(x,0) = v(x),\] by a dissipative Galerkin method. When v is taken to have a jump discontinuity at zero, that discontinuity will propagate along $x + t = 0$, in the true solution u. Estimates in ${L_2}$ and ${L_\infty }$ of the pollution effects of the discontinuity are found. These estimates show those effects to decay exponentially in ${h^{ - 1}}$ in regions a fixed distance d from the discontinuity and exponentially in d for fixed h.