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Preprint 07-2013

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

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Author(s) : Maciek Korzec , Tobias Ahnert

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 07-2013

MSC 2000

65M70 Spectral, collocation and related methods
65L06 Multistep, Runge-Kutta and extrapolation methods

Abstract :
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.

Keywords : Quantum dot self-assembly, time-stepping, GPU computing, IMEX, pseudospectral method, convexity splitting, exponential integrators, MATLAB

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