High-order numerical methods for 2D parabolic problems in single and composite domains

In this work, we discuss and compare three methods for the numerical approximation of constant- and variable-coefficient diffusion equations in both single and composite domains with possible discontinuity in the solution/flux at interfaces, considering (i) the Cut Finite Element Method; (ii) the Difference Potentials Method; and (iii) the summation-by-parts Finite Difference Method. First we give a brief introduction for each of the three methods. Next, we propose benchmark problems, and consider numerical tests-with respect to accuracy and convergence-for linear parabolic problems on a single domain, and continue with similar tests for linear parabolic problems on a composite domain (with the interface defined either explicitly or implicitly). Lastly, a comparative discussion of the methods and numerical results will be given.Comment: 45 pages, 12 figures, in revision for Journal of Scientific Computin

Colloidal ionic complexes on periodic substrates: ground state configurations and pattern switching

We theoretically and numerically studied ordering of "colloidal ionic clusters" on periodic substrate potentials as those generated by optical trapping. Each cluster consists of three charged spherical colloids: two negatively and one positively charged. The substrate is a square or rectangular array of traps, each confining one such cluster. By varying the lattice constant from large to small, the observed clusters are first rod-like and form ferro- and antiferro-like phases, then they bend into a banana-like shape and finally condense into a percolated structure. Remarkably, in a broad parameter range between single-cluster and percolated structures, we have found stable supercomplexes composed of six colloids forming grape-like or rocket-like structures. We investigated the possibility of macroscopic pattern switching by applying external electrical fields.Comment: 14 pages, 13 figure

A study of blow-ups in the Keller-Segel model of chemotaxis

We study the Keller-Segel model of chemotaxis and develop a composite particle-grid numerical method with adaptive time stepping which allows us to accurately resolve singular solutions. The numerical findings (in two dimensions) are then compared with analytical predictions regarding formation and interaction of singularities obtained via analysis of the stochastic differential equations associated with the Keller-Segel model