106,224 research outputs found
Application of the operator splitting to the Maxwell equations with the source term
Motivated by numerical solution of the time-dependent Maxwell equations, we consider splitting methods for a linear system of differential equations split into two subproblems and First, expressions for the leading term of the local error are derived for the Strang-Marchuk and the symmetrically weighted sequential splitting methods. The analysis, done in assumption that the subproblems are solved exactly, confirms the expected second order global accuracy of both schemes. Second, several relevant numerical tests are performed for the Maxwell equations discretized in space either by finite differences or by finite elements. An interesting case is the splitting into the subproblems and (with the split-off source term ). For the central finite difference staggered discretization, we consider second order splitting schemes and compare them to the classical Yee scheme on a test problem with loss and source terms. For the vector Nédélec finite element discretizations, we test the Gautschi-Krylov time integration scheme. Applied in combination with the split-off source term, it leads to splitting schemes that are exact per split step. Thus, the time integration error of the schemes consists solely of the splitting error
Oscillation-free method for semilinear diffusion equations under noisy initial conditions
Noise in initial conditions from measurement errors can create unwanted
oscillations which propagate in numerical solutions. We present a technique of
prohibiting such oscillation errors when solving initial-boundary-value
problems of semilinear diffusion equations. Symmetric Strang splitting is
applied to the equation for solving the linear diffusion and nonlinear
remainder separately. An oscillation-free scheme is developed for overcoming
any oscillatory behavior when numerically solving the linear diffusion portion.
To demonstrate the ills of stable oscillations, we compare our method using a
weighted implicit Euler scheme to the Crank-Nicolson method. The
oscillation-free feature and stability of our method are analyzed through a
local linearization. The accuracy of our oscillation-free method is proved and
its usefulness is further verified through solving a Fisher-type equation where
oscillation-free solutions are successfully produced in spite of random errors
in the initial conditions.Comment: 19 pages, 9 figure
Splitting and composition methods in the numerical integration of differential equations
We provide a comprehensive survey of splitting and composition methods for
the numerical integration of ordinary differential equations (ODEs). Splitting
methods constitute an appropriate choice when the vector field associated with
the ODE can be decomposed into several pieces and each of them is integrable.
This class of integrators are explicit, simple to implement and preserve
structural properties of the system. In consequence, they are specially useful
in geometric numerical integration. In addition, the numerical solution
obtained by splitting schemes can be seen as the exact solution to a perturbed
system of ODEs possessing the same geometric properties as the original system.
This backward error interpretation has direct implications for the qualitative
behavior of the numerical solution as well as for the error propagation along
time. Closely connected with splitting integrators are composition methods. We
analyze the order conditions required by a method to achieve a given order and
summarize the different families of schemes one can find in the literature.
Finally, we illustrate the main features of splitting and composition methods
on several numerical examples arising from applications.Comment: Review paper; 56 pages, 6 figures, 8 table
Order reduction methods for solving large-scale differential matrix Riccati equations
We consider the numerical solution of large-scale symmetric differential
matrix Riccati equations. Under certain hypotheses on the data, reduced order
methods have recently arisen as a promising class of solution strategies, by
forming low-rank approximations to the sought after solution at selected
timesteps. We show that great computational and memory savings are obtained by
a reduction process onto rational Krylov subspaces, as opposed to current
approaches. By specifically addressing the solution of the reduced differential
equation and reliable stopping criteria, we are able to obtain accurate final
approximations at low memory and computational requirements. This is obtained
by employing a two-phase strategy that separately enhances the accuracy of the
algebraic approximation and the time integration. The new method allows us to
numerically solve much larger problems than in the current literature.
Numerical experiments on benchmark problems illustrate the effectiveness of the
procedure with respect to existing solvers
Palindromic 3-stage splitting integrators, a roadmap
The implementation of multi-stage splitting integrators is essentially the
same as the implementation of the familiar Strang/Verlet method. Therefore
multi-stage formulas may be easily incorporated into software that now uses the
Strang/Verlet integrator. We study in detail the two-parameter family of
palindromic, three-stage splitting formulas and identify choices of parameters
that may outperform the Strang/Verlet method. One of these choices leads to a
method of effective order four suitable to integrate in time some partial
differential equations. Other choices may be seen as perturbations of the
Strang method that increase efficiency in molecular dynamics simulations and in
Hybrid Monte Carlo sampling.Comment: 20 pages, 8 figures, 2 table
Operator splittings and spatial approximations for evolution equations
The convergence of various operator splitting procedures, such as the
sequential, the Strang and the weighted splitting, is investigated in the
presence of a spatial approximation. To this end a variant of Chernoff's
product formula is proved. The methods are applied to abstract partial delay
differential equations.Comment: to appear in J. Evol. Equations. Reviewers comments are incorporate
The role of numerical boundary procedures in the stability of perfectly matched layers
In this paper we address the temporal energy growth associated with numerical
approximations of the perfectly matched layer (PML) for Maxwell's equations in
first order form. In the literature, several studies have shown that a
numerical method which is stable in the absence of the PML can become unstable
when the PML is introduced. We demonstrate in this paper that this instability
can be directly related to numerical treatment of boundary conditions in the
PML. First, at the continuous level, we establish the stability of the constant
coefficient initial boundary value problem for the PML. To enable the
construction of stable numerical boundary procedures, we derive energy
estimates for the variable coefficient PML. Second, we develop a high order
accurate and stable numerical approximation for the PML using
summation--by--parts finite difference operators to approximate spatial
derivatives and weak enforcement of boundary conditions using penalties. By
constructing analogous discrete energy estimates we show discrete stability and
convergence of the numerical method. Numerical experiments verify the
theoretical result
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