196 research outputs found
Discrete line integral method for the Lorentz force system
In this paper, we apply the Boole discrete line integral to solve the Lorentz
force system which is written as a non-canonical Hamiltonian system. The method
is exactly energy-conserving for polynomial Hamiltonians of degree . In any other case, the energy can also be conserved approximatively. With
comparison to well-used Boris method, numerical experiments are presented to
demonstrate the energy-preserving property of the method
Numerical Implementation of the Multisymplectic Preissman Scheme and Its Equivalent Schemes
We analyze the multisymplectic Preissman scheme for the KdV equation with the
periodic boundary condition and show that the unconvergence of the widely-used
iterative methods to solve the resulting nonlinear algebra system of the
Preissman scheme is due to the introduced potential function. A artificial
numerical condition is added to the periodic boundary condition. The added
boundary condition makes the numerical implementation of the multisymplectic
Preissman scheme practical and is proved not to change the numerical solutions
of the KdV equation. Based on our analysis, we derive some new schemes which
are not restricted by the artificial boundary condition and more efficient than
the Preissman scheme because of less computing cost and less computer storages.
By eliminating the auxiliary variables, we also derive two schemes for the KdV
equation, one is a 12-point scheme and the other is an 8-point scheme. As the
byproducts, we present two new explicit schemes which are not multisymplectic
but still have remarkable numerical stable property.Comment: 21page
Optimal error estimate of two linear and momentum-preserving Fourier pseudo-spectral schemes for the RLW equation
In this paper, two novel linear-implicit and momentum-preserving Fourier
pseudo-spectral schemes are proposed and analyzed for the regularized long-wave
equation. The numerical methods are based on the blend of the Fourier
pseudo-spectral method in space and the linear-implicit Crank-Nicolson method
or the leap-frog scheme in time. The two fully discrete linear schemes are
shown to possess the discrete momentum conservation law, and the linear systems
resulting from the schemes are proved uniquely solvable. Due to the momentum
conservative property of the proposed schemes, the Fourier pseudo-spectral
solution is proved to be bounded in the discrete norm. Then by
using the standard energy method, both the linear-implicit Crank-Nicolson
momentum-preserving scheme and the linear-implicit leap-frog
momentum-preserving scheme are shown to have the accuracy of
in the discrete norm without any
restrictions on the grid ratio, where is the number of nodes and is
the time step size. Numerical examples are carried out to verify the correction
of the theory analysis and the efficiency of the proposed schemes
Structure-preserving algorithms for the two-dimensional sine-Gordon equation with Neumann boundary conditions
This paper presents two kinds of strategies to construct structure-preserving
algorithms with homogeneous Neumann boundary conditions for the sine-Gordon
equation, while most existing structure-preserving algorithms are only valid
for zero or periodic boundary conditions. The first strategy is based on the
conventional second-order central difference quotient but with a cell-centered
grid, while the other is established on the regular grid but incorporated with
summation by parts (SBP) operators. Both the methodologies can provide
conservative semi-discretizations with different forms of Hamiltonian
structures and the discrete energy. However, utilizing the existing SBP
formulas, schemes obtained by the second strategy can directly achieve
higher-order accuracy while it is not obvious for schemes based on the
cell-centered grid to make accuracy improved easily. Further combining the
symplectic Runge-Kutta method and the scalar auxiliary variable (SAV) approach,
we construct symplectic integrators and linearly implicit energy-preserving
schemes for the two-dimensional sine-Gordon equation, respectively. Extensive
numerical experiments demonstrate their effectiveness with the homogeneous
Neumann boundary conditions.Comment: 23 pages, 47 figure
Partitioned AVF methods
The classic second-order average vector field (AVF) method can exactly
preserve the energy for Hamiltonian ordinary differential equations and partial
differential equations. However, the AVF method inevitably leads to
fully-implicit nonlinear algebraic equations for general nonlinear systems. To
address this drawback and maintain the desired energy-preserving property, a
first-order partitioned AVF method is proposed which first divides the
variables into groups and then applies the AVF method step by step. In
conjunction with its adjoint method we present the partitioned AVF composition
method and plus method respectively to improve its accuracy to second order.
Concrete schemes for two classic model equations are constructed with
semi-implicit, linear-implicit properties that make considerable lower cost
than the original AVF method. Furthermore, additional conservative property can
be generated besides the conventional energy preservation for specific
problems. Numerical verification of these schemes further conforms our results.Comment: 23 pages, 20 figure
A linearized and conservative Fourier pseudo-spectral method for the damped nonlinear Schr\"{o}dinger equation in three dimensions
In this paper, we propose a linearized Fourier pseudo-spectral method, which
preserves the total mass and energy conservation laws, for the damped nonlinear
Schr\"{o}dinger equation in three dimensions. With the aid of the semi-norm
equivalence between the Fourier pseudo-spectral method and the finite
difference method, an optimal -error estimate for the proposed method
without any restriction on the grid ratio is established by analyzing the real
and imaginary parts of the error function. Numerical results are addressed to
confirm our theoretical analysis.Comment: 29 pages, 2 figure
A Novel Sixth Order Energy-Conserved Method for Three-Dimensional Time-Domain Maxwell's Equations
In this paper, a novel sixth order energy-conserved method is proposed for
solving the three-dimensional time-domain Maxwell's equations. The new scheme
preserves five discrete energy conservation laws, three momentum conservation
laws, symplectic conservation law as well as two divergence-free properties and
is proved to be unconditionally stable, non-dissipative. An optimal error
estimate is established based on the energy method, which shows that the
proposed method is of sixth order accuracy in time and spectral accuracy in
space in discrete -norm. The constant in the error estimate is proved to
be only . Furthermore, the numerical dispersion relation is analyzed in
detail and a fast solver is presented to solve the resulting discrete linear
equations efficiently. Numerical results are addressed to verify our
theoretical analysis.Comment: 36 page
A novel linearized and momentum-preserving Fourier pseudo-spectral scheme for the Rosenau-Korteweg de Vries equation
In this paper, we design a novel linearized and momentum-preserving Fourier
pseudo-spectral scheme to solve the Rosenau-Korteweg de Vries equation. With
the aid of a new semi-norm equivalence between the Fourier pseudo-spectral
method and the finite difference method, a prior bound of the numerical
solution in discrete -norm is obtained from the discrete momentum
conservation law. Subsequently, based on the energy method and the bound of the
numerical solution, we show that, without any restriction on the mesh ratio,
the scheme is convergent with order in discrete
-norm, where is the number of collocation points used in the
spectral method and is the time step. Numerical results are addressed to
confirm our theoretical analysis.Comment: 24 pages.arXiv admin note: text overlap with arXiv:1808.0685
A linearly implicit energy-preserving exponential integrator for the nonlinear Klein-Gordon equation
In this paper, we generalize the exponential energy-preserving integrator
proposed in the recent paper [SIAM J. Sci. Comput. 38(2016) A1876-A1895] for
conservative systems, which now becomes linearly implicit by further utilizing
the idea of the scalar auxiliary variable approach. Comparing with the original
exponential energy-preserving integrator which usually leads to a nonlinear
algebraic system, our new method only involve a linear system with constant
coefficient matrix. Taking the nonlinear Klein-Gordon equation for example, we
derive the concrete energy-preserving scheme and demonstrate its high
efficiency through numerical experiments.Comment: 21 pages, 13 figure
A linearly implicit structure-preserving scheme for the fractional sine-Gordon equation based on the IEQ approach
This paper aims to develop a linearly implicit structure-preserving numerical
scheme for the space fractional sine-Gordon equation, which is based on the
newly developed invariant energy quadratization method. First, we reformulate
the equation as a canonical Hamiltonian system by virtue of the variational
derivative of the functional with fractional Laplacian. Then, we utilize the
fractional centered difference formula to discrete the equivalent system
derived by the invariant energy quadratization method in space direction, and
obtain a conservative semi-discrete scheme. Subsequently, the linearly implicit
structure-preserving method is applied for the resulting semi-discrete system
to arrive at a fully-discrete conservative scheme. The stability, solvability
and convergence in the maximum norm of the numerical scheme are given.
Furthermore, a fast algorithm based on the fast Fourier transformation
technique is used to reduce the computational complexity in practical
computation. Finally, numerical examples are provided to confirm our
theoretical analysis results.Comment: 28pages, 5 figure
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