D. C. Antonopoulos

D. c. Antonopoulos, V. a. Dougalis, G. Kounadis

We consider a simple initial-boundary-value problem for the shallow water equations in one space dimension. We discretize the problem in space by the standard Galerkin finite element method on a quasiuniform mesh and in time by the classical 4-stage, 4th order, explicit Runge-Kutta scheme. Assuming smoothness of solutions, a Courant number restriction, and certain hypotheses on the finite element spaces, we prove L2 error estimates that are of fourth-order accuracy in the temporal variable and of the usual, due to the nonuniform mesh, suboptimal order in space. We also make a computational study of the numerical spatial and temporal orders of convergence, and of the validity of a hypothesis made on the finite element spaces.

D. C. Antonopoulos, V. A. Dougalis

We consider the Shallow Water equations in the supercritical and subcritical cases in one space variable,posed in a finite spatial interval with characteristic boundary conditions at the endpoints, which, as is well known, are transparent,i.e. allow outgoing waves to exit without generating spurious reflected waves. Assuming that the resulting initial-boundary-value problems have smooth solutions,we approximate them in space using standard Galerkin-finite element methods and prove L^2 error estimates for the semidiscrete problems on quasiuniform meshes.We discretize the problems in the temporal variable using an explicit,fourth-order accurate Runge-Kutta scheme and check, by means of numerical experiment, that the resulting fully discrete schemes have excellent absorption properties.

D. C. Antonopoulos, V. A. Dougalis

We consider the `classical' Boussinesq system in one space dimension and its symmetric analog. These systems model two-way propagation of nonlinear, dispersive long waves of small amplitude on the surface of an ideal fluid in a uniform horizontal channel. We discretize an initial-boundary-value problem for these systems in space using Galerkin-finite element methods and prove error estimates for the resulting semidiscrete problems and also for their fully discrete analogs effected by explicit Runge-Kutta time-stepping procedures. The theoretical orders of convergence obtained are consistent with the results of numerical experiments that are also presented.

D. C. Antonopoulos, V. A. Dougalis, D. E. Mitsotakis

We consider the Camassa-Holm (CH) equation, a nonlinear dispersive wave equation that models one-way propagation of long waves of moderately small amplitude. We discretize in space the periodic initial-value problem for CH (written in its original and in system form), using the standard Galerkin finite element method with smooth splines on a uniform mesh, and prove optimal-order $L^{2}$-error estimates for the semidiscrete approximation. We also consider an initial-boundary-value problem on a finite interval for the system form of CH and analyze the convergence of its standard Galerkin semidiscretization. Using the fourth-order accurate, explicit, "classical" Runge-Kutta scheme for time-stepping, we construct a highly accurate, stable, fully discrete scheme that we employ in numerical experiments to approximate solutions of CH, mainly smooth travelling waves and nonsmooth solitons of the `peakon' type.

D. C. Antonopoulos, V. A. Dougalis

We consider a simple initial-boundary-value problem for the shallow water equations in one space dimension, and also the analogous problem for a symmetric variant of the system. Assuming smoothness of solutions, we discretize these problems in space using standard Galerkin-finite element methods and prove $L^{2}$-error estimates for the semidiscrete problems for quasiuniform and uniform meshes. In particular we show that in the case of spatial discretizations with piecewise linear continuous functions on a uniform mesh, suitable compatibility conditions at the boundary and superaccuracy properties of the $L^{2}$ projection on the finite element subspaces lead to an optimal-order $O(h^{2})$ $L^{2}$-error estimate. We also examine temporal discretizations of the semidiscrete problems by three explicit Runge-Kutta methods (the Euler, improved Euler, and the Shu-Osher scheme) and prove $L^{2}$-error estimates, which are of optimal order in the temporal variable, under appropriate stability conditions. In a final section of remarks we prove optimal-order $L^{2}$-error estimates for smooth spline spatial discretizations of the periodic initial-value problem for the systems. We also prove that small-amplitude, appropriately transformed solutions of the symmetric system are close to the corresponding solutions of the usual system while they are both smooth, thus providing a justification of the symmetric system.