Nonlinear scheme with high accuracy for nonlinear coupled parabolic-hyperbolic system
The authors construct an unconditionally stable nonlinear three-level scheme with second-order \(H^1\)- and \(L^2\)-norm accuracy in temporal and spatial variables, respectively, for two-dimensional nonlinear parabolic-hyperbolic systems. The stability and convergence properties of this scheme are rigorously studied. Difficulties arising from the nonlinearity and coupling between parabolic and hyperbolic equations are overcome by an ingenious use of the method of energy estimation and inductive hypothesis reasoning. This method differs from those used for linear schemes and can be adopted to perform strict theoretical analysis of other nonlinear schemes for nonlinear partial differential equation problems. Numerical tests verify the theoretical results.
scientific article; zbMATH DE number 5895480