Shock viscosities are used in explicit finite element calculations to eliminate the oscillations that occur behind wave fronts. A multidimensional shock viscosity based on the total variation diminishing (TVD) methods in computational fluid dynamics is presented. Example calculations demonstrate that it is superior to the traditional shock viscosity formulations for problems involving both strong and weak shocks. Furthermore, we find that: the shock viscosity is relatively insensitive to the choice of the TVD limiter, equals or betters the previous flux-limited viscosity, and achieves a nearly minimal shock width of two elements in strong shock calculations.