Notes on error estimates for the standard Galerkin-finite element method for the Shallow Water equations

2014/03/22 by Dimitrios Antonopoulos, Antonopoulos, D. C., Vassilios A. Dougalis +1 · 2 citations
Engineering · Mathematics · #35L60 (secondary) #65M60 (primary) #Advanced Numerical Methods in Computational Mathematics #Applied mathematics #Backward Euler method #Boundary value problem #Computational Fluid Dynamics and Aerodynamics #Differential Equations and Numerical Methods #Discretization #Euler equations #FOS: Mathematics #Finite element method #Galerkin method #Mathematical analysis #Mathematics #Numerical Analysis (math.NA) #Piecewise #Shallow water equations

paper · pdf · doi:10.48550/arxiv.1403.5699

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2014/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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 L²-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 L² projection on the finite element subspaces lead to an optimal-order O(h²) L²-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 L²-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 L²-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.

Citations

• Boussinesq Equations and Other Systems for Small-Amplitude Long Waves in Nonlinear Dispersive Media. I: Derivation and Linear Theory
• Error Estimates to Smooth Solutions of Runge--Kutta Discontinuous Galerkin Methods for Scalar Conservation Laws
• Error Estimates to Smooth Solutions of Runge–Kutta Discontinuous Galerkin Method for Symmetrizable Systems of Conservation Laws
• Stability Analysis and A Priori Error Estimates of the Third Order Explicit Runge–Kutta Discontinuous Galerkin Method for Scalar Conservation Laws

Cited by

• Notes on Galerkin-finite element methods for the Shallow Water equations with characteristic boundary conditions

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