52 research outputs found

    Solving periodic semilinear stiff PDEs in 1D, 2D and 3D with exponential integrators

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    Dozens of exponential integration formulas have been proposed for the high-accuracy solution of stiff PDEs such as the Allen-Cahn, Korteweg-de Vries and Ginzburg-Landau equations. We report the results of extensive comparisons in MATLAB and Chebfun of such formulas in 1D, 2D and 3D, focusing on fourth and higher order methods, and periodic semilinear stiff PDEs with constant coefficients. Our conclusion is that it is hard to do much better than one of the simplest of these formulas, the ETDRK4 scheme of Cox and Matthews

    LeXInt: Package for Exponential Integrators employing Leja interpolation

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    We present a publicly available software for exponential integrators that computes the φl(z)\varphi_l(z) functions using polynomial interpolation. The interpolation method at Leja points have recently been shown to be competitive with the traditionally-used Krylov subspace method. The developed framework facilitates easy adaptation into any Python software package for time integration.Comment: Publicly available software available at https://github.com/Pranab-JD/LeXInt, in submissio

    An exponential integration generalized multiscale finite element method for parabolic problems

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    We consider linear and semilinear parabolic problems posed in high-contrast multiscale media in two dimensions. The presence of high-contrast multiscale media adversely affects the accuracy, stability, and overall efficiency of numerical approximations such as finite elements in space combined with some time integrator. In many cases, implementing time discretizations such as finite differences or exponential integrators may be impractical because each time iteration needs the computation of matrix operators involving very large and ill-conditioned sparse matrices. Here, we propose an efficient Generalized Multiscale Finite Element Method (GMsFEM) that is robust against the high-contrast diffusion coefficient. We combine GMsFEM with exponential integration in time to obtain a good approximation of the final time solution. Our approach is efficient and practical because it computes matrix functions of small matrices given by the GMsFEM method. We present representative numerical experiments that show the advantages of combining exponential integration and GMsFEM approximations. The constructions and methods developed here can be easily adapted to three-dimensional domains

    Time-stepping methods for the simulation of the self-assembly of nano-crystals in MATLAB on a GPU

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    Partial differential equations describing the patterning of thin crystalline films are typically of fourth or sixth order, they are quasi- or semilinear and they are mostly defined on simple geometries such as rectangular domains. For the numerical simulation of these kind of problems spectral methods are an efficient approach. We apply several implicit-explicit schemes to one recently derived PDE that we express in terms of coefficients of trigonometric interpolants. While the simplest IMEX scheme turns out to have the mildest step-size restriction, higher order SBDF schemes tend to be more unstable and exponential time integrators are fastest for the calculation of very accurate solutions. We implemented a reduced model in the EXPINT package syntax and compared various exponential schemes. A convexity splitting approach was employed to stabilize the SBDF1 scheme. We show that accuracy control is crucial when using this idea, therefore we present a time-adaptive SBDF1/SBDF1-2-step method that yields convincing results reflecting the change in timescales during topological changes of the nanostructures. The implementation of all presented methods is carried out in MATLAB. We used the open source GPUmat package to gain up to 5-fold runtime benefits by carrying out calculations on a low-cost GPU without having to prescribe any knowledge in low-level programming or CUDA implementations and found comparable speedups as with MATLAB's PCT or with GPUmat run on Octave

    Equivalence between the DPG method and the Exponential Integrators for linear parabolic problems

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    The Discontinuous Petrov-Galerkin (DPG) method and the exponential integrators are two well established numerical methods for solving Partial Di fferential Equations (PDEs) and sti ff systems of Ordinary Di fferential Equations (ODEs), respectively. In this work, weapply the DPG method in the time variable for linear parabolic problems and we calculate the optimal test functions analytically. We show that the DPG method in time is equivalent to exponential integrators for the trace variables, which are decoupled from the interior variables. In addition, the DPG optimal test functions allow us to compute the approximated solutions in the time element interiors. This DPG method in time allows to construct a posteriori error estimations in order to perform adaptivity. We generalize this novel DPG-based time-marching scheme to general fi rst order linear systems of ODEs. We show the performance of the proposed method for 1D and 2D + time linear parabolic PDEs after discretizing in space by the nite element method

    Direction splitting of φ\varphi-functions in exponential integrators for dd-dimensional problems in Kronecker form

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    In this manuscript, we propose an efficient, practical and easy-to-implement way to approximate actions of φ\varphi-functions for matrices with dd-dimensional Kronecker sum structure in the context of exponential integrators up to second order. The method is based on a direction splitting of the involved matrix functions, which lets us exploit the highly efficient level 3 BLAS for the actual computation of the required actions in a μ\mu-mode fashion. The approach has been successfully tested on two- and three-dimensional problems with various exponential integrators, resulting in a consistent speedup with respect to a technique designed to compute actions of φ\varphi-functions for Kronecker sums

    Conservation of phase space properties using exponential integrators on the cubic Schrödinger equation

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    The cubic nonlinear Schrödinger (NLS) equation with periodic boundary conditions is solvable using Inverse Spectral Theory. The nonlinear spectrum of the associated Lax pair reveals topological properties of the NLS phase space that are difficult to assess by other means. In this paper we use the invariance of the nonlinear spectrum to examine the long time behavior of exponential and multisymplectic integrators as compared with the most commonly used split step approach. The initial condition used is a perturbation of the unstable plane wave solution, which is difficult to numerically resolve. Our findings indicate that the exponential integrators from the viewpoint of efficiency and speed have an edge over split step, while a lower order multisymplectic is not as accurate and too slow to compete. © 2006 Elsevier Inc. All rights reserved

    A DPG-based time-marching scheme for linear hyperbolic problems

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    The Discontinuous Petrov-Galerkin (DPG) method is a widely employed discretization method for Partial Di fferential Equations (PDEs). In a recent work, we applied the DPG method with optimal test functions for the time integration of transient parabolic PDEs. We showed that the resulting DPG-based time-marching scheme is equivalent to exponential integrators for the trace variables. In this work, we extend the aforementioned method to time-dependent hyperbolic PDEs. For that, we reduce the second order system in time to first order and we calculate the optimal testing analytically. We also relate our method with exponential integrators of Gautschi-type. Finally, we validate our method for 1D/2D + time linear wave equation after semidiscretization in space with a standard Bubnov-Galerkin method. The presented DPG-based time integrator provides expressions for the solution in the element interiors in addition to those on the traces. This allows to design di fferent error estimators to perform adaptivity
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