TY - JOUR
TI - On the standard Galerkin method with explicit RK4 time stepping for the shallow water equations
AU - Antonopoulos, D.C.
AU - Dougalis, V.A.
AU - Kounadis, G.
JO - IMA Journal of Numerical Analysis
PY - 2020
VL - 40
TODO - 4
SP - 2415-2449
PB - Oxford University Press
SN - 0272-4979, 1464-3642
TODO - 10.1093/IMANUM/DRZ033
TODO - 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
TODO - 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.
ER -