Error estimates for the standard Galerkin-finite element method for the shallow water equations

Abstract

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 2error 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.

Galerkin finite element methods for the Shallow Water equations over variable bottom

“…Using the approximation and inverse properties of S h and S h,0 , we may then estimate the various terms in the r.h.s. of (2.10) for t ∈ [0, t h ] in a similar way as in [10], since β ∈ C 1 , and conclude that for t ∈ [0, t h ]…”

“…In the case of a uniform mesh it is expected that the L 2 errors of the semidiscrete solution will be of O(h r ) while, for a quasiuniform mesh, of O(h r−1 ), cf. [10]. We verified these rates of accuracy in numerical experiments using C 0 linear, C 2 cubic and C 4 quintic splines (i.e.…”

Discrete energy estimates for the $abcd$-systems

“…A systematic method to construct appropriate semidiscrete schemes is presented in [32]. More recently, this case has been studied in [5]. The authors construct a finite-volume-Galerkin numerical scheme, where discrete estimates in L 2 × L 2 are obtained.…”

Galerkin-finite element methods for the shallow water equations with characteristic boundary conditions

“…[D1] for the analysis in the case of a linear model problem. In [AD2] it was proved that the order of convergence in L 2 for piecewise linear continuous elements on a uniform mesh is equal to 2 in the case of an ibvp for (1.1) with the homogeneous boundary conditions u(0, t) = u(L, t) = 0. This superaccuracy result is expected to hold for the ibvp's under consideration as well and this is indeed what the numerical experiments of section 4 indicate.)…”

“…(H1)-(H3) in section 2. The proof also requires that r ≥ 3 so that a certain bootstrap argument, based on the boundedness of the · 1,∞ norm of an error term, goes through as in [D2], [AD2].…”