862 research outputs found
Hamiltonian interpolation of splitting approximations for nonlinear PDEs
We consider a wide class of semi linear Hamiltonian partial differential
equa- tions and their approximation by time splitting methods. We assume that
the nonlinearity is polynomial, and that the numerical tra jectory remains at
least uni- formly integrable with respect to an eigenbasis of the linear
operator (typically the Fourier basis). We show the existence of a modified
interpolated Hamiltonian equation whose exact solution coincides with the
discrete flow at each time step over a long time depending on a non resonance
condition satisfied by the stepsize. We introduce a class of modified splitting
schemes fulfilling this condition at a high order and prove for them that the
numerical flow and the continuous flow remain close over exponentially long
time with respect to the step size. For stan- dard splitting or
implicit-explicit scheme, such a backward error analysis result holds true on a
time depending on a cut-off condition in the high frequencies (CFL condition).
This analysis is valid in the case where the linear operator has a discrete
(bounded domain) or continuous (the whole space) spectrum
Hamiltonian interpolation of splitting approximations for nonlinear PDEs
International audienceWe consider a wide class of semi linear Hamiltonian partial differential equa- tions and their approximation by time splitting methods. We assume that the nonlinearity is polynomial, and that the numerical tra jectory remains at least uni- formly integrable with respect to an eigenbasis of the linear operator (typically the Fourier basis). We show the existence of a modified interpolated Hamiltonian equation whose exact solution coincides with the discrete flow at each time step over a long time depending on a non resonance condition satisfied by the stepsize. We introduce a class of modified splitting schemes fulfilling this condition at a high order and prove for them that the numerical flow and the continuous flow remain close over exponentially long time with respect to the step size. For stan- dard splitting or implicit-explicit scheme, such a backward error analysis result holds true on a time depending on a cut-off condition in the high frequencies (CFL condition). This analysis is valid in the case where the linear operator has a discrete (bounded domain) or continuous (the whole space) spectrum
An approximation scheme for semilinear parabolic PDEs with convex and coercive Hamiltonians
We propose an approximation scheme for a class of semilinear parabolic
equations that are convex and coercive in their gradients. Such equations arise
often in pricing and portfolio management in incomplete markets and, more
broadly, are directly connected to the representation of solutions to backward
stochastic differential equations. The proposed scheme is based on splitting
the equation in two parts, the first corresponding to a linear parabolic
equation and the second to a Hamilton-Jacobi equation. The solutions of these
two equations are approximated using, respectively, the Feynman-Kac and the
Hopf-Lax formulae. We establish the convergence of the scheme and determine the
convergence rate, combining Krylov's shaking coefficients technique and
Barles-Jakobsen's optimal switching approximation.Comment: 24 page
Energy preserving model order reduction of the nonlinear Schr\"odinger equation
An energy preserving reduced order model is developed for two dimensional
nonlinear Schr\"odinger equation (NLSE) with plane wave solutions and with an
external potential. The NLSE is discretized in space by the symmetric interior
penalty discontinuous Galerkin (SIPG) method. The resulting system of
Hamiltonian ordinary differential equations are integrated in time by the
energy preserving average vector field (AVF) method. The mass and energy
preserving reduced order model (ROM) is constructed by proper orthogonal
decomposition (POD) Galerkin projection. The nonlinearities are computed for
the ROM efficiently by discrete empirical interpolation method (DEIM) and
dynamic mode decomposition (DMD). Preservation of the semi-discrete energy and
mass are shown for the full order model (FOM) and for the ROM which ensures the
long term stability of the solutions. Numerical simulations illustrate the
preservation of the energy and mass in the reduced order model for the two
dimensional NLSE with and without the external potential. The POD-DMD makes a
remarkable improvement in computational speed-up over the POD-DEIM. Both
methods approximate accurately the FOM, whereas POD-DEIM is more accurate than
the POD-DMD
An operator splitting scheme for the fractional kinetic Fokker-Planck equation
In this paper, we develop an operator splitting scheme for the fractional
kinetic Fokker-Planck equation (FKFPE). The scheme consists of two phases: a
fractional diffusion phase and a kinetic transport phase. The first phase is
solved exactly using the convolution operator while the second one is solved
approximately using a variational scheme that minimizes an energy functional
with respect to a certain Kantorovich optimal transport cost functional. We
prove the convergence of the scheme to a weak solution to FKFPE. As a
by-product of our analysis, we also establish a variational formulation for a
kinetic transport equation that is relevant in the second phase. Finally, we
discuss some extensions of our analysis to more complex systems
Exponentially accurate Hamiltonian embeddings of symplectic A-stable Runge--Kutta methods for Hamiltonian semilinear evolution equations
We prove that a class of A-stable symplectic Runge--Kutta time
semidiscretizations (including the Gauss--Legendre methods) applied to a class
of semilinear Hamiltonian PDEs which are well-posed on spaces of analytic
functions with analytic initial data can be embedded into a modified
Hamiltonian flow up to an exponentially small error. As a consequence, such
time-semidiscretizations conserve the modified Hamiltonian up to an
exponentially small error. The modified Hamiltonian is -close to the
original energy where is the order of the method and the time
step-size. Examples of such systems are the semilinear wave equation or the
nonlinear Schr\"odinger equation with analytic nonlinearity and periodic
boundary conditions. Standard Hamiltonian interpolation results do not apply
here because of the occurrence of unbounded operators in the construction of
the modified vector field. This loss of regularity in the construction can be
taken care of by projecting the PDE to a subspace where the operators occurring
in the evolution equation are bounded and by coupling the number of excited
modes as well as the number of terms in the expansion of the modified vector
field with the step size. This way we obtain exponential estimates of the form
with and ; for the semilinear wave
equation, , and for the nonlinear Schr\"odinger equation, . We give
an example which shows that analyticity of the initial data is necessary to
obtain exponential estimates
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