diffusion equation

We solve the phase-field equations in two dimensions to simulate crystal growth in the low undercooling regime. The novelty is the use of a fast solver for the free space heat equation to compute the thermal field. This solver is based on the efficient direct evaluation of the integral representation of the solution to the constant coefficient, free space heat equation with a smooth source term. The computational cost and memory requirements of the new solver are reasonable and no artificial boundary conditions are needed. This allows one to solve for the thermal field in a computational domain whose size depends only on the size of the growing crystal and not on the extent of the thermal field, which can result in significant computational savings in the low undercooling regime.

The mathematical formulation of mass transfer in drying processes is often based on the nonlinear unsteady diffusion equation. In general, numerical simulations are required to solve these equations. Very often, however, indirect and simplified methods neglecting fundamentals of the processes are used. In this work, a new mathematical model approach for the mass transfer occurring during drying of sliced foods is proposed. The model considers fundamentals of the drying process and takes internal resistance to moisture transfer into account. The parameters in the formulation have physical meaning and permit giving clear view of the moisture depletion process occurring during drying. The proposed model has an analytical solution and allows finding effective diffusion coefficient accurately. The verification of the model is made with basic drying experiments performed for chili red peppers sliced in slab form. The results reveal that there is nearly perfect match between the drying curves obtained by the model and the experiments.

This paper presents a diffusion method for generating terrains from a set of parameterized curves that characterize the landform features such as ridge lines, riverbeds or cliffs. Our approach provides the user with an intuitive vector-based feature-oriented control over the terrain. Different types of constraints (such as elevation, slope angle and roughness) can be attached to the curves so as to define the shape of the terrain. The terrain is generated from the curve representation by using an efficient multigrid diffusion algorithm. The algorithm can be efficiently implemented on the GPU, which allows the user to interactively create a vast variety of landscapes.

We compute u(t) = exp(-tA)ϕ using rational Krylov subspace reduction for 0 ≤ t < ∞, where u(t), ϕ ∈ R N and 0 < A = A * ∈ R N×N . A priori optimization of the rational Krylov subspaces for this problem was considered in [V. Druskin, L. Knizhnerman, and M. Zaslavsky, SIAM J. Sci. Comput., 31 (2009), pp. 3760-3780]. There was suggested an algorithm generating sequences of equidistributed shifts, which are asymptotically optimal for the cases with uniform spectral distributions. Here we develop a recursive greedy algorithm for choice of shifts taking into account nonuniformity of the spectrum. The algorithm is based on an explicit formula for the residual in the frequency domain allowing adaptive shift optimization at negligible cost. The effectiveness of the developed approach is demonstrated in an example of the three-dimensional diffusion problem for Maxwell's equation arising in geophysical exploration. We compare our approach with the one using the above-mentioned equidistributed sequences of shifts. Numerical examples show that our algorithm is able to adapt to the spectral density of operator A. For examples with near-uniform spectral distributions, both algorithms show the same convergence rates, but the new algorithm produces superior convergence for cases with nonuniform spectra.

This study presents a numerical scheme for solving one dimensional equations of conservation law form. The Saulyev's finite difference techniques are used to compute the solution. Although the resulting difference equation do not appear explicit, a suitable use of the equation make it explicit. It is shown that this explicit scheme is unconditionally stable. A numerical example is presented to demonstrate the accuracy and efficiency of the proposed computational procedure.

The numerical solution of partial differential equations with finite differences mimetic methods that satisfy properties of the continuum differential operators and mimic discrete versions of appropriate integral identities is more likely to produce better approximations. Recently, one of the authors developed a systematic approach to obtain mimetic finite difference discretizations for divergence and gradient operators, which achieves the same order of accuracy on the boundary and inner grid points. This paper uses the second-order version of those operators to develop a new mimetic finite difference method for the steady-state diffusion equation. A complete theoretical and numerical analysis of this new method is presented, including an original and nonstandard proof of the quadratic convergence rate of this new method. The numerical results agree in all cases with our theoretical analysis, providing strong evidence that the new method is a better choice than the standard finite difference method.

... [5,11] Liou and Bruin were developed short-cut drying theory to describe the drying process and solve the nonlinear differential equation in modelling. ... In the opposite of Neumann boundary condition, the mass flux at the interface is variable in Robin boundary condition. ...

The numerical solution of two-dimensional, linear and non-linear elliptic partial differential equations (PDEs) using two parallel algorithms namely Lawrie Sameh and domain decomposition has been computed on three parallel architectures. The PDEs considered here, describe the diffusion of a pollutant released from a constant source, variable source and a point source. In addition, an example with diffusion and chemical removal modelled by a non-linear PDE and three-dimensional diffusion equation is given. The parallel algorithms have been implemented on three-different systems (i) SUNFIRE (ii) IBM and (iii) PARAM. The scalability analysis has been done to analyze the performance of both the algorithms on all three parallel systems.

Model nonlinear diffusion equation with the most simple Landau-Ginzburg free energy functional was applied to locate boundaries between meaningful regions of low-level images. The method is oriented to processing images of objects that are a result of dynamic evolution: images of different organs and tissues obtained by radiography and NMR methods, electron microscope images of morphogenesis fields, etc. In the

A diffuse-interface model is considered for solving axisymmetric immiscible twophase flow with surface tension. The Navier-Stokes (NS) equations are modified by the addition of a continuum forcing. The interface between the two fluids is considered as the half level set of a mass concentration c, which is governed by the Cahn-Hilliard (CH) equation-a fourth order, degenerate, nonlinear parabolic diffusion equation. In this work, we develop a nonlinear multigrid method to solve the CH equation with degenerate mobility and couple this to a projection method for the incompressible NS equations. The diffuse-interface method can deal with topological transitions such as breakup and coalescence smoothly without ad hoc Ôcut and connectÕ or other artificial procedures. We present results for RayleighÕs capillary instability up to forming satellite drops. The results agree well with the linear stability theory.

This paper proposes a new quality improvement technique for fractal-based image compression techniques using diffusion equations. Fractal coding uses a contractive mapping scheme to represent an image. This process of contractive mapping causes artifacts and blocking effects in encoded images. This problem is severed when compression ratio is increased or there are high frequency regions in the image. Hence, to

Vast though the literature on the chemistry of the alkali ± silica reaction (ASR) in concrete has become, a comprehensive mathematical model allowing quantitative predictions seems lacking. The present study attempts a step toward this goal. While two distinct problems must be dealt with, namely, (1) the kinetics of the chemical reaction with the associated diffusion processes and (2) fracture mechanics of the damage process, only the former is addressed here. The analysis is focused on the recent attempts by C. Meyers and W. Jin to incorporate ground waste glass (mainly, bottle glass) into concrete. With minor adjustments, though, the model can be applied to ASR in natural aggregates as well. A characteristic unit cubic cell of concrete containing one spherical glass particle is analyzed. A spherical layer of basic ASR gel grows radially inward into the particle, controlled by diffusion of water toward the reaction front. Modification of the solution for the case of mineral aggregates with veins of silica is also indicated. Imbibition of additional water from the adjacent capillary pores, which causes swelling of the gel, is described as a second diffusion process, limited by the development of pressure due to resistance of concrete to expansion. The water used up to form the basic ASR gel and imbibed to cause its swelling appears as a sink term in the non-linear diffusion equation for the global water transport through a concrete structure. The differential equations are integrated numerically. The study of the effects of various parameters provides improved understanding of the ASR, and especially the effect of glass particle size. Full prediction will require measurements of some parameters of the reaction processes.

The performance of conjugate gradient (CG) algorithms for the solution of the system of linear equations that results from the finite-differencing of the neutron diffusion equation was analyzed on SIMD, MIMD, and mixed-mode parallel machines. A block preconditioner based on the incomplete Cholesky factorization was used to accelerate the conjugate gradient search. The issues involved in mapping both the unpreconditioned