Matthew A. Beauregard

An adaptive splitting approach for the quenching solution of reaction-diffusion equations over nonuniform grids

The numerical solution of a nonlinear degenerate reaction-diffusion equation of the quenching type is investigated. While spatial derivatives are discretized over symmetric nonuniform meshes, a Peaceman-Rachford splitting method is employed to advance solutions of the semidiscretized system. The temporal step is determined adaptively through a suitable arc-length monitor function. A criterion is derived to ensure that the numerical solution acquired preserves correctly the positivity and monotonicity of the analytical solution. Weak stability is proven in a von Neumann sense via the ~-norm. Computational examples are presented to illustrate our results.

A semi-adaptive compact splitting method for the numerical solution of 2-dimensional quenching problems

Abstract This article studies a semi-adaptive compact Peaceman–Rachford splitting method for solving two-dimensional nonlinear reaction–diffusion equations with singular source terms. While an adaption is utilized in the temporal direction, uniform grids are considered in the space. It is shown that the compact scheme is stable and convergent when its dimensional Courant numbers are within the frame of a window determined by the given spatial domain. Though such a window implies a considerable restriction on the decomposed compact scheme, the new computational strategy acquired is highly efficient and reliable for a variety of applications. Numerical examples are given to illustrate our conclusions and indicate that the method constructed is effective in determining key singularity characteristics such as the quenching time and critical domain.

Casimir effect in the presence of external fields

In this work the Casimir effect is studied for scalar fields in the presence of boundaries and under the influence of arbitrary smooth potentials of compact support. In this setting, piston configurations are analyzed in which the piston is modeled by a potential. For these configurations, analytic results for the Casimir energy and force are obtained by employing the zeta function regularization method. Also, explicit numerical results for the Casimir force are provided for pistons modeled by a class of compactly supported potentials that are realizable as delta-sequences. These results are then generalized to higher dimensional pistons by considering additional Kaluza–Klein dimensions.

Casimir energies in spherically symmetric background potentials revisited

In this paper we reconsider the formulation for the computation of the Casimir energy in spherically symmetric background potentials. Compared to the previous analysis, the technicalities are much easier to handle and final answers are surprisingly simple.

Improving the speed of convergence of GMRES for certain perturbed tridiagonal systems

Numerical approximations of partial differential equations often require the employment of spatial adaptation or the utilization of non-uniform grids to resolve fine details of the solution. While the governing continuous linear operator may be symmetric, the discretized version may lose this essential property as a result of adaptation or utilization of non-uniform grids. Commonly, the matrices can be viewed as a perturbation to a known matrix or to a previous iterates matrix. In either case, a linear solver is deployed to solve the resulting linear system. Iterative methods provide a plausible and affordable way of completing this task and Krylov subspace methods, such as GMRES, are quite popular. Upon updating the matrices as a result of adaptation or multi-grid methodologies, approximate eigenvector information is known stemming from the prior GMRES iterative method. Hence, this information can be utilized to improve the convergence rate of the subsequent iterative method. A one dimensional Poisson problem is examined to illustrate this methodology while showing notable and quantifiable improvements over standard methods, such as GMRES-DR.

Explorations and Expectations of Equidistribution Adaptations for Nonlinear Quenching Problems

Finite difference computations that involve spatial adaptation commonly employ an equidistribution principle. In these cases, a new mesh is constructed such that a given monitor function is equidistributed in some sense. Typical choices of the monitor function involve the solution or one of its many derivatives. This straightforward concept has proven to be extremely effective and practical. However, selections of core monitoring functions are often challenging and crucial to the computational success. This paper concerns six different designs of the monitoring function that targets a highly nonlinear partial differential equation that exhibits both quenching-type and degeneracy singularities. While the first four monitoring strategies are within the so-called primitive regime, the rest belong to a later category of the modified type, which requires the priori knowledge of certain important quenching solution characteristics. Simulated examples are given to illustrate our study and conclusions.

A nonlinear splitting algorithm for systems of partial differential equations with self-diffusion

Systems of reactiondiffusion equations are commonly used in biological models of food chains. The populations and their complicated interactions present numerous challenges in theory and in numerical approximation. In particular, self-diffusion is a nonlinear term that models overcrowding of a particular species. The nonlinearity complicates attempts to construct efficient and accurate numerical approximations of the underlying systems of equations. In this paper, a new nonlinear splitting algorithm is designed for a partial differential equation that incorporates self-diffusion. We present a general model that incorporates self-diffusion and develop a numerical approximation. The numerical analysis of the approximation provides criteria for stability and convergence. Numerical examples are used to illustrate the theoretical results.

Compact schemes in application to singular reaction-diffusion equations

A high order compact scheme is employed to obtain the numerical solution of a singular, one-dimensional, reaction-diffusion equation of the quenching-type motivated by models describing combustion processes. The adaptation of the temporal step is discussed in light of the proposed theory. A condition, reminiscent of the Courant-Friedrichs-Lewy (CFL) condition, is determined to guarantee that the numerical solution monotonically increases, a property the analytic solution is known to exhibit. Strong stability is proven in a Von-Neumann sense via the 2-norm. Computational examples illustrate the spatial convergence and quenching times are calculated for particular singular source terms.

A variable nonlinear splitting algorithm for reaction diffusion systems with self- and cross- diffusion

Self- and cross-diffusion are important nonlinear spatial derivative terms that are included into biological models of predator-prey interactions. Self-diffusion models overcrowding effects, while cross-diffusion incorporates the response of one species in light of the concentration of another. In this paper, a novel nonlinear operator splitting method is presented that directly incorporates both self- and cross-diffusion into a computational efficient design. The numerical analysis guarantees the accuracy and demonstrates appropriate criteria for stability. Numerical experiments display its efficiency and accuracy.

Numerical solutions to singular reaction-diffusion equation over elliptical domains

Solid fuel ignition models, for which the dynamics of the temperature is independent of the single-species mass fraction, attempt to follow the dynamics of an explosive event. Such models may take the form of singular, degenerate, reaction-diffusion equations of the quenching type, that is, the temporal derivative blows up in finite time while the solution remains bounded. Theoretical and numerical investigations have proved difficult for even the simplest of geometries and mathematical degeneracies. Quenching domains are known to exist for piecewise smooth boundaries, but often lack theoretical estimates. Rectangular geometries have been primarily studied. Here, this acquired knowledge is utilized to determine new theoretical estimates to quenching domains for arbitrary piecewise, smooth, connected geometries. Elliptical domains are of primary interest and a Peaceman-Rachford splitting algorithm is then developed that employs temporal adaptation and nonuniform grids. Rigorous numerical analysis ensures numerical solution monotonicity, positivity, and linear stability of the proposed algorithm. Simulation experiments are provided to illustrate the accomplishments.