arXiv subjects

Kevin T. Chu

Publications and source records attributed to Kevin T. Chu.

7 recordsLinked to original sources

arXiv2009

Boosting the Accuracy of Finite Difference Schemes via Optimal Time Step Selection and Non-Iterative Defect Correction

In this article, we present a simple technique for boosting the order of accuracy of finite difference schemes for time dependent partial differential equations by optimally selecting the time step used to advance the numerical solution and adding defect correction terms in a non-iterative manner. The power of the technique is its ability to extract as much accuracy as possible from existing finite difference schemes with minimal additional effort. Through straightforward numerical analysis arguments, we explain the origin of the boost in accuracy and estimate the computational cost of the resulting numerical method. We demonstrate the utility of optimal time step (OTS) selection combined with non-iterative defect correction (NIDC) on several different types of finite difference schemes for a wide array of classical linear and semilinear PDEs in one and more space dimensions on both regular and irregular domains.

math.NA↗

arXiv2007

Surface Conservation Laws at Microscopically Diffuse Interfaces

In studies of interfaces with dynamic chemical composition, bulk and interfacial quantities coupled via surface conservation laws of excess surface quantities. While this approach is for microscopically sharp interfaces, its applicability in the context of microscopically diffuse is less theoretically well-established. Furthermore, surface conservation laws (and interfacial in general) are often derived phenomenologically rather than systematically. In this article, provide a mathematically rigorous justification for surface conservation laws at diffuse interfaces on an asymptotic analysis of transport processes in the boundary layer and derive general the surface and normal fluxes that appear in surface conservation laws. Next, we use non-thermodynamics to formulate surface conservation laws in terms of chemical potentials a method for systematically deriving the structure of the interfacial layer. Finally, we conservation laws for a few examples from diffusive and electrochemical transport.

physics.chem-ph↗

arXiv2007

A Direct Matrix Method for Computing Analytical Jacobians of Discretized Nonlinear Integro-differential Equations

In this pedagogical article, we present a simple direct matrix method for analytically computing the Jacobian of nonlinear algebraic equations that arise from the discretization of nonlinear integro-differential equations. The method is based on a formulation of the discretized equations in vector form using only matrix-vector products and component-wise operations. By applying simple matrix-based differentiation rules, the matrix form of the analytical Jacobian can be calculated with little more difficulty than that required when computing derivatives in single-variable calculus. After describing the direct matrix method, we present numerical experiments demonstrating the computational performance of the method, discuss its connection to the Newton-Kantorovich method, and apply it to illustrative 1D and 2D example problems. MATLAB code is provided to demonstrate the low code complexity required by the method.

math.NA↗

arXiv2006

A Variational Level Set Approach for Surface Area Minimization of Triply Periodic Surfaces

In this paper, we study triply periodic surfaces with minimal surface area under a constraint in the volume fraction of the regions (phases) that the surface separates. Using a variational level set method formulation, we present a theoretical characterization of and a numerical algorithm for computing these surfaces. We use our theoretical and computational formulation to study the optimality of the Schwartz P, Schwartz D, and Schoen G surfaces when the volume fractions of the two phases are equal and explore the properties of optimal structures when the volume fractions of the two phases not equal. Due to the computational cost of the fully, three-dimensional shape optimization problem, we implement our numerical simulations using a parallel level set method software package.

math.OC↗

arXiv2006

Nonlinear electrochemical relaxation around conductors

We analyze the simplest problem of electrochemical relaxation in more than one dimension - the response of an uncharged, ideally polarizable metallic sphere (or cylinder) in a symmetric, binary electrolyte to a uniform electric field. In order to go beyond the circuit approximation for thin double layers, our analysis is based on the Poisson-Nernst-Planck (PNP) equations of dilute solution theory. Unlike most previous studies, however, we focus on the nonlinear regime, where the applied voltage across the conductor is larger than the thermal voltage. In such strong electric fields, the classical model predicts that the double layer adsorbs enough ions to produce bulk concentration gradients and surface conduction. Our analysis begins with a general derivation of surface conservation laws in the thin double-layer limit, which provide effective boundary conditions on the quasi-neutral bulk. We solve the resulting nonlinear partial differential equations numerically for strong fields and also perform a time-dependent asymptotic analysis for weaker fields, where bulk diffusion and surface conduction arise as first-order corrections. We also derive various dimensionless parameters comparing surface to bulk transport processes, which generalize the Bikerman-Dukhin number. Our results have basic relevance for double-layer charging dynamics and nonlinear electrokinetics in the ubiquitous PNP approximation.

cond-mat.mtrl-sci↗

arXiv2004

Current-Voltage Relations for Electrochemical Thin Films

The dc response of an electrochemical thin film, such as the separator in a micro-battery, is analyzed by solving the Poisson-Nernst-Planck equations, subject to boundary conditions appropriate for an electrolytic/galvanic cell. The model system consists of a binary electrolyte between parallel-plate electrodes, each possessing a compact Stern layer, which mediates Faradaic reactions with nonlinear Butler-Volmer kinetics. Analytical results are obtained by matched asymptotic expansions in the limit of thin double layers and compared with full numerical solutions. The analysis shows that (i) decreasing the system size relative to the Debye screening length decreases the voltage of the cell and allows currents higher than the classical diffusion-limited current; (ii) finite reaction rates lead to the important possibility of a reaction-limited current; (iii) the Stern-layer capacitance is critical for allowing the cell to achieve currents above the reaction-limited current; and (iv) all polarographic (current-voltage) curves tend to the same limit as reaction kinetics become fast. Dimensional analysis, however, shows that ``fast'' reactions tend to become ``slow'' with decreasing system size, so the nonlinear effects of surface polarization may dominate the dc response of thin films.

physics.chem-ph↗

arXiv2004

Electrochemical Thin Films At and Above the Classical Limiting Current

We study a model electrochemical thin film at dc currents exceeding the classical diffusion-limited value. The mathematical problem involves the steady Poisson-Nernst-Planck equations for a binary electrolyte with nonlinear boundary conditions for reaction kinetics and Stern-layer capacitance, as well as an integral constraint on the number of anions. At the limiting current, we find a nested boundary layer structure at the cathode, which is required by the reaction boundary condition. Above the limiting current, a depletion of anions generally characterizes the cathode side of the cell. In this regime, we derive leading-order asymptotic approximations for the (i) classical bulk space-charge layer and (ii) another, nested highly charged boundary layer at the cathode. The former involves an exact solution to the Nernst-Planck equations for a single, unscreened ionic species, which may apply more generally to Faradaic conduction through very thin insulating films. By matching expansions, we derive current-voltage relations well into the space-charge regime. Throughout our analysis, we emphasize the strong influence of the Stern-layer capacitance on cell behavior.

physics.chem-ph↗