Numerical methods for nonlinear second-order hyperbolic partial differential equations. II – Rothe’s techniques for 1-D problems

Two families of Rothe’s methods that are based on the discretization of the time variable and keeping the spatial one continuous for the solution of second-order hyperbolic equations with damping and nonlinear source terms are presented. The first family is based on time integration and results in a Volterra integro-differential equation which, upon approximating certain time integrals, can be written as an ordinary differential equation in space. Upon discretizing the spatial derivatives by means of finite difference formula, this family results in nonlinear algebraic equations which contain exponential terms that depend on the time step, and its accuracy is shown to be lower than that of an explicit second-order accurate discretization in both space and time of second-order hyperbolic equations. The second family of methods is based on the discretization of the time derivatives and time linearization, and results in linear ordinary differential equations in space. Upon freezing the coefficients of these equations, one can integrate analytically the resulting linear ordinary differential equations to obtain piecewise exponential solutions which are continuous in space, and three-point finite difference equations which depend exponentially on the time step, spatial grid size, and the diffusion, relaxation, damping and reaction times. The finite difference equations are shown to result in non-diagonally dominant matrices unless the time step is smaller than the characteristic relaxation, diffusion, damping and reaction times. To avoid this problem, two Rothe’s techniques that do not account for the Jacobian of the reaction terms in the ordinary differential operator are developed, and it is shown that the Rothe’s techniques belonging to the second family are as accurate as the linearly implicit methods presented in Part I provided that the relaxation time is smaller than the critical time and the solution does not oscillate in space. When the relaxation time is greater than the critical one, Rothe’s methods have been found to be less accurate than the finite difference techniques of Part I. The Rothe’s techniques belonging to the second family are generalized to systems of nonlinear second-order hyperbolic equations and mixed systems of parabolic and second-order hyperbolic partial differential equations.