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The new numerical scheme based on the convex splitting method allows for large time step sizes.
For large time scales, the PDFs are Gaussian for both data sets.
We obtain a formula for the asymptotic behaviour of the Dirichlet heat kernel for large time in terms of the survival probability of a Brownian motion, under the assumption that the latter decays subexponentially for large time.
Our tests show that the proposed solver maintains its optimal efficiency at high Reynolds numbers and for large time steps.
Consequently, the Markov semigroup converges exponentially to the invariant probability measure in entropy and is compact for large time.
For large time simulations, we also introduce a methodology based on the asymptotic expansion of coefficients involved in exact direct transparent boundary conditions.
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And to reduce the time cost for large time-consuming complex structure optimization, parallel computing is built-in in the MATLAB codes.
Vartools is designed especially for batch processing of light curves, including built-in support for parallel processing, making it useful for large time-domain surveys such as searches for transiting planets.
Results show that the numerical solution converges to the exact solution for large times.
Finally, we find the asymptotics for large times t of solutions u x, t) of the time-dependent Dirac equation.
The asymptotic value of this coefficient for large times approaches the value predicted by the Taylor dispersion theory.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com