We introduce cuPentBatch -our own pentadiagonal solver for NVIDIA GPUs. The development of cuPentBatch has been motivated by applications involving numerical solutions of parabolic partial differential equations, which we describe. Our solver is written with batch processing in mind (as necessitated by parameter studies of various physical models). In particular, our solver is directed at those problems where only the right-hand side of the matrix changes as the batch solutions are generated. As such, we demonstrate that cuPentBatch outperforms the NVIDIA standard pentadiagonal batch solver gpsvInterleavedBatch for the class of physically-relevant computational problems encountered herein. Program SummaryProgram Title: cuPentBatch https://github.com/munstermonster/cuPentBatch Licensing Provision: Apache License 2.0

We introduce a mathematical model with a mesh-free numerical method to describe contact-line motion in lubrication theory. We show how the model resolves the singularity at the contact line, and generates smooth profiles for an evolving, spreading droplet. The model describes well the physics of droplet spreading–including Tanner’s Law for the evolution of the contact line. The model can be configured to describe complete wetting or partial wetting, and we explore both cases numerically. In the case of partial wetting, the model also admits analytical solutions for the droplet profile, which we present here. Article highlights We formulate a mathematical model to regularize the contact-line singularity for droplet spreading. The model can be solved using a fast, accurate mesh-free numerical method. Numerical simulations confirm that the model describes the quantitative aspects of droplet spreading well.

We develop a mathematical framework for determining the stability of steady states of generic nonlinear reaction-diffusion equations with periodic source terms, in one spatial dimension. We formulate an a priori condition for the stability of such steady states, which relies only on the properties of the steady state itself. The mathematical framework is based on Bloch's theorem and Poincaré's inequality for mean-zero periodic functions. Our framework can be used for stability analysis to determine the regions in an appropriate parameter space for which steady-state solutions are stable.

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