@article{3063385,
    title = "On the standard Galerkin method with explicit RK4 time stepping for the shallow water equations",
    author = "Antonopoulos, D.C. and Dougalis, V.A. and Kounadis, G.",
    journal = "IMA Journal of Numerical Analysis",
    year = "2020",
    volume = "40",
    number = "4",
    pages = "2415-2449",
    publisher = "Oxford University Press",
    issn = "0272-4979, 1464-3642",
    doi = "10.1093/IMANUM/DRZ033",
    keywords = "Finite element method;  Galerkin methods;  Initial value problems;  Mesh generation;  Runge Kutta methods, Error estimates;  Explicit Runge-Kutta methods;  Finite element space;  Fourth order four-stage explicit runge-kuttum method;  Fourth-order;  Galerkin finite element methods;  Shallow water equations;  Simple++;  Standard galerkin finite element method;  Time-stepping, Equations of motion",
    abstract = "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 four-stage, fourth 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. © 2020 Oxford University Press. All rights reserved."
}