Related papers: Numerical preservation of multiple local conservat…
We use the general framework of summation-by-parts operators to construct conservative, energy-stable, and well-balanced semidiscretizations of two different nonlinear systems of dispersive shallow water equations with varying bathymetry:…
We present the multiplier method of constructing conservative finite difference schemes for ordinary and partial differential equations. Given a system of differential equations possessing conservation laws, our approach is based on…
A key consideration in the development of numerical schemes for time-dependent partial differential equations (PDEs) is the ability to preserve certain properties of the continuum solution, such as associated conservation laws or other…
Invariant finite-difference schemes are considered for one-dimensional magnetohydrodynamics (MHD) equations in mass Lagrangian coordinates for the cases of finite and infinite conductivity. For construction these schemes previously obtained…
The compact finite difference method is a powerful tool for discretizing conservation laws, owing to its inherent flexibility in developing high-resolution and highly stable schemes. In this paper, we propose a framework for the design of…
A method for constructing first integral preserving numerical schemes for time-dependent partial differential equations on non-uniform grids is presented. The method can be used with both finite difference and partition of unity approaches,…
In the second part of this series, we use the Lagrange multiplier approach proposed in the first part \cite{CheS21} to construct efficient and accurate bound and/or mass preserving schemes for a class of semi-linear and quasi-linear…
We propose a structure-preserving finite difference scheme for the Cahn-Hilliard equation with a dynamic boundary condition using the discrete variational derivative method (DVDM). In this approach, it is important and essential how to…
The one-dimensional modified shallow water equations in Lagrangian coordinates are considered. It is shown the relationship between symmetries and conservation laws in Lagrangian coordinates, in mass Lagrangian variables, and Eulerian…
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…
An effective algorithmic method is presented for finding the local conservation laws for partial differential equations with any number of independent and dependent variables. The method does not require the use or existence of a…
A method based on infinite parameter conservation laws is described to factor linear differential operators out of nonlinear partial differential equations (PDEs) or out of differential consequences of nonlinear PDEs. This includes a…
We present direct methods and symbolic software for the computation of conservation laws of nonlinear partial differential equations (PDEs) and differential-difference equations (DDEs).The methods are applied to nonlinear PDEs in (1+1)…
Using the method of equivariant moving frames, we present a procedure for constructing symmetry-preserving finite element methods for second-order ordinary differential equations. Using the method of lines, we then indicate how our…
This paper gives a general treatment and proof of the direct conservation law method presented in Part I. In particular, the treatment here applies to finding the local conservation laws of any system of one or more partial differential…
In this paper, we introduce a Lagrange multiplier approach to construct linearly implicit energy-preserving schemes of arbitrary order for general Hamiltonian PDEs. Unlike the widely used auxiliary variable methods, this novel approach does…
In this work we propose a new, arbitrary order space-time finite element discretisation for Hamiltonian PDEs in multisymplectic formulation. We show that the new method which is obtained by using both continuous and discontinuous…
We present linearly implicit methods that preserve discrete approximations to local and global energy conservation laws for multi-symplectic PDEs with cubic invariants. The methods are tested on the one-dimensional Korteweg-de Vries…
For partial differential equations (PDEs) that have $n\geq2$ independent variables and a symmetry algebra of dimension at least $n-1$, an explicit algorithmic method is presented for finding all symmetry-invariant conservation laws that…
Differential equations arising in fluid mechanics are usually derived from the intrinsic properties of mechanical systems, in the form of conservation laws, and bear symmetries, which are not generally preserved by a finite difference…