TY - JOUR
T1 - A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations
AU - Ranocha , Hendrik
AU - Mitsotakis , Dimitrios
AU - I. Ketcheson , David
JO - Communications in Computational Physics
VL - 4
SP - 979
EP - 1029
PY - 2021
DA - 2021/02
SN - 29
DO - http://doi.org/10.4208/cicp.OA-2020-0119
UR - https://global-sci.org/intro/article_detail/cicp/18643.html
KW - Invariant conservation, summation by parts, finite difference methods, Galerkin
methods, relaxation schemes.
AB - <p style="text-align: justify;">We develop a general framework for designing conservative numerical
methods based on summation by parts operators and split forms in space, combined
with relaxation Runge-Kutta methods in time. We apply this framework to create
new classes of fully-discrete conservative methods for several nonlinear dispersive
wave equations: Benjamin-Bona-Mahony (BBM), Fornberg-Whitham, Camassa-Holm,
Degasperis-Procesi, Holm-Hone, and the BBM-BBM system. These full discretizations
conserve all linear invariants and one nonlinear invariant for each system. The spatial
semidiscretizations include finite difference, spectral collocation, and both discontinuous and continuous finite element methods. The time discretization is essentially
explicit, using relaxation Runge-Kutta methods. We implement some specific schemes
from among the derived classes, and demonstrate their favorable properties through
numerical tests.</p>