Time evolution of discrete fourth-order elliptic operators

The evolution equation (Formula presented.) is considered. A discrete parabolic methodology is developed, based on a discrete elliptic (fourth-order) calculus. The main ingredient of this calculus is a discrete biharmonic operator (DBO). In the general case, it is shown that the approximate solutions converge to the continuous one. An “almost optimal” convergence result (O(h ^{4 − ϵ} )) is established in the case of constant coefficients, in particular in the pure biharmonic case. Several numerical test cases are presented that not only corroborate the theoretical accuracy result, but also demonstrate high-order accuracy of the method in nonlinear cases. The nonlinear equations include the well-studied Kuramoto–Sivashinsky equation. Numerical solutions for this equation are shown to approximate remarkably well the exact solutions. The numerical examples demonstrate the great improvement achieved by using the DBO instead of the standard (five-point) discrete bilaplacian.