Finite difference method

Abstract: A new numerical technique by Chang is described and used to solve the one-dimensional (1D)g and two-dimensional (2D) Saint Venant equations. This new technique differs from traditional numerical methods (i.e., finite-difference, finite-element, finite-volume, spectral methods, etc.). Chang’s method treats space and time on the same footing, so that space and time are unified—key characteristic that distinguishes the new method from other techniques. This method is explicit, uses a staggered grid, enforces flux conservation in space and time, and does not require upwinding, flux-splitting, flux limiters, evaluation of eigenvalues, or the addition of artificial viscosity. Furthermore, the scheme is simple, easy to implement (see Appendix I), and can be extended to higher dimensions. First, the new technique, as developed by Chang, is introduced, explained, and applied to the 1D Saint Venant equations. To illustrate its effectiveness, an idealized dam-break and a hydraulic jump in a straight rectangular channel are simulated. The numerical results obtained using the new method are compared with results from other, more conventional techniques of similar complexity, experimental data, and an analytical solution. Next, Chang’s scheme is extended for solution of the 2D Saint Venant (i.e., shallow water or depth-averaged) equations. The writers do not follow Chang’s 2D development but instead employ Strang’s method of fractional steps, which transforms the 2D problem into two 1D problems. The new 2D scheme is applied to an oblique hydraulic jump, and the results are compared with an analytical solution.

Abstract: We obtain a sufficient condition for the existence of periodic and subharmonic solutions of second-order p-Laplacian difference equations using the critical point theory.