The hopscotch method is a numerical integration method that can be applied efficiently to the system of ordinary differential equations arising by space-discretization of time dependent partial differential equations (PDEs). The error propagation of this method from one time level to another can be described by a three-term matrix-vector recursion. In regards to PDEs that relate to (multi-dimensional) bending beam equations, we obtain an explicit sharp bound for the discrete L₂-norm of the propagated error. This bound is expressed in quantities determined by spectral properties of the matrices in the recursion. The expression leads to conditions under which the hopscotch method is strongly asymptotically stable: that is, the stability is uniform with respect to the mesh widths in space and time (that correlate and that tend to 0) and to the time level (that tends to :3wk). In many applications, our conditions completely describe the situation in which the hopscotch method is strongly asymptotically stable. Previously known stability conditions are more restrictive and do not give complete descriptions.

--Author’s Abstract

The stability analysis for the fourth-order bending beam equation is sufficiently general to provide a corresponding analysis for second-order parabolic problems such as diffusion-convection equations.