A difference method for plane problems in dymanic elasticity

The equations governing dynamic elastic deformation under conditions of plane strain are written as a system of symmetric, hyperbolic, first order, partial differential equations with constant coefficients. A system of explicit difference equations with second order accuracy are presented for the solution of mixed initial and boundary value problems for regions composed of rectangles. At interior points the difference scheme is the same as the scheme proposed by Lax and Wendroff. They established the schemes conditional stability for initial value problems. At boundary points Butler’s procedure based on integration along bicharacteristics is used to derive the appropriate difference equations. The method is applied to a problem for which the exact solution is known. Numerical evidence indicates that the method is stable for a mesh ratio $k/h$ approximately twice as large as required by the Lax—Wendroff condition. The error in total energy is $O\left ( {{k^3}} \right )$ and increases linearly with increasing $t$.