Solution of binary multiphase diffusion problems allowing for variable diffusivity, with application to the aluminization of nickel
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Sarkhel, A. K.↗
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First-Order Hyperbolic System Method for Time-Dependent Advection-Diffusion Problems
A time-dependent extension of the first-order hyperbolic system method for advection-diffusion problems is introduced. Diffusive/viscous terms are written and discretized as a hyperbolic system, which recovers the original equation in the steady state. The resulting scheme offers advantages over traditional schemes: a dramatic simplification in the discretization, high-order accuracy in the solution gradients, and orders-of-magnitude convergence acceleration. The hyperbolic advection-diffusion system is discretized by the second-order upwind residual-distribution scheme in a unified manner, and the system of implicit-residual-equations is solved by Newton's method over every physical time step. The numerical results are presented for linear and nonlinear advection-diffusion problems, demonstrating solutions and gradients produced to the same order of accuracy, with rapid convergence over each physical time step, typically less than five Newton iterations.
Mazaheri, Alireza↗
Existence, uniqueness and regularity of a time-periodic probability density distribution arising in a sedimentation-diffusion problem
The sedimentation and diffusion of a nonneutrally buoyant Brownian particle in vertical fluid-filled cylinder of finite length which is instantaneously inverted at regular intervals are investigated analytically. A one-dimensional convective-diffusive equation is derived to describe the temporal and spatial evolution of the probability density; a periodicity condition is formulated; the applicability of Fredholm theory is established; and the parameter-space regions are determined within which the existence and uniqueness of solutions are guaranteed. Numerical results for sample problems are presented graphically and briefly characterized.
Nitsche, Ludwig C.↗
Ternary phase diagrams for the Ga-P-Zn and Ga-As-Zn systems with applications to diffusion problems
Ternary phase diagrams for gallium-phosphorus-zinc and gallium-arsenic-zinc systems with diffusion problem applications
ZINC↗
Multigrid approaches to non-linear diffusion problems on unstructured meshes
The efficiency of three multigrid methods for solving highly non-linear diffusion problems on two-dimensional unstructured meshes is examined. The three multigrid methods differ mainly in the manner in which the nonlinearities of the governing equations are handled. These comprise a non-linear full approximation storage (FAS) multigrid method which is used to solve the non-linear equations directly, a linear multigrid method which is used to solve the linear system arising from a Newton linearization of the non-linear system, and a hybrid scheme which is based on a non-linear FAS multigrid scheme, but employs a linear solver on each level as a smoother. Results indicate that all methods are equally effective at converging the non-linear residual in a given number of grid sweeps, but that the linear solver is more efficient in cpu time due to the lower cost of linear versus non-linear grid sweeps.
Mavriplis, Dimitri J.↗
An integral solution to a nonlinear diffusion problem
The generalized Galerkin method (or the method of integral relations) is applied to the type of problem described by quasilinear parabolic equations. As an example the problem of nonlinear transient slab diffusion with a general reservoir boundary condition is worked out. The integral relations are given for an arbitrary number of strips, and solutions using up to seven strips have been obtained in order to investigate the convergence of the method.
Hussaini, M. Y.↗
Application of the space-time conservation element and solution element method to two-dimensional advection-diffusion problems
The existing 2-D alpha-mu scheme and alpha-epsilon scheme based on the method of space-time conservation element and solution element, which were constructed for solving the linear 2-D unsteady advection-diffusion equation and unsteady advection equation, respectively, are tested. Also, the alpha-epsilon scheme is modified to become the V-E scheme for solving the nonlinear 2-D inviscid Burgers equation. Numerical solutions of six test problems are presented in comparison with their exact solutions or numerical solutions obtained by traditional finite-difference or finite-element methods. It is demonstrated that the 2-D alpha-mu, alpha-epsilon, and nu-epsilon schemes can be used to obtain numerical results which are more accurate than those based on some of the traditional methods but without using any artificial tuning in the computation. Similar to the previous 1-D test problems, the high accuracy and simplicity features of the space-time conservation element and solution element method have been revealed again in the present 2-D test results.
Wang, Xiao-Yen↗
Application of the Space-Time Conservation Element and Solution Element Method to One-Dimensional Advection-Diffusion Problems
Test problems are used to examine the performance of several one-dimensional numerical schemes based on the space-time conservation and solution element (CE/SE) method. Investigated in this paper are the CE/SE schemes constructed previously for solving the linear unsteady advection-diffusion equation and the schemes derived here for solving the nonlinear viscous and inviscid Burgers equations. In comparison with the numerical solutions obtained using several traditional finite-difference schemes with similar accuracy, the CE/SE solutions display much lower numerical dissipation and dispersion errors.
Wang, Xiao-Yen↗
Extrapolation techniques applied to matrix methods in neutron diffusion problems
A general matrix method is developed for the solution of characteristic-value problems of the type arising in many physical applications. The scheme employed is essentially that of Gauss and Seidel with appropriate modifications needed to make it applicable to characteristic-value problems. An iterative procedure produces a sequence of estimates to the answer; and extrapolation techniques, based upon previous behavior of iterants, are utilized in speeding convergence. Theoretically sound limits are placed on the magnitude of the extrapolation that may be tolerated. This matrix method is applied to the problem of finding criticality and neutron fluxes in a nuclear reactor with control rods. The two-dimensional finite-difference approximation to the two-group neutron fluxes in a nuclear reactor with control rods. The two-dimensional finite-difference approximation to the two-group neutron-diffusion equations is treated. Results for this example are indicated.
Mccready, Robert R↗
Higher order methods for convection-diffusion problems
This paper applies C1 cubic Hermite polynomials embedded in an orthogonal collocation scheme to the spatial discretization of the unsteady nonlinear Burgers equation as a model of the equations of fluid mechanics. The temporal discretization is carried out by means of either a noniterative finite difference or an iterative finite difference procedure. Results of this method are compared with those of a second-order finite difference scheme and a splined-cubic Taylor's series scheme. Stability limits are derived and the matrix structure of the several schemes are compared.
Murphy, J. D.↗
An adaptive pseudo-spectral method for reaction diffusion problems
The spectral interpolation error was considered for both the Chebyshev pseudo-spectral and Galerkin approximations. A family of functionals I sub r (u), with the property that the maximum norm of the error is bounded by I sub r (u)/J sub r, where r is an integer and J is the degree of the polynomial approximation, was developed. These functionals are used in the adaptive procedure whereby the problem is dynamically transformed to minimize I sub r (u). The number of collocation points is then chosen to maintain a prescribed error bound. The method is illustrated by various examples from combustion problems in one and two dimensions.
Bayliss, A.↗
An adaptive pseudo-spectral method for reaction diffusion problems
The spectral interpolation error was considered for both the Chebyshev pseudo-spectral and Galerkin approximations. A family of functionals I sub r (u), with the property that the maximum norm of the error is bounded by I sub r (u)/J sub r, where r is an integer and J is the degree of the polynomial approximation, was developed. These functionals are used in the adaptive procedure whereby the problem is dynamically transformed to minimize I sub r (u). The number of collocation points is then chosen to maintain a prescribed error bound. The method is illustrated by various examples from combustion problems in one and two dimensions.
Bayliss, A.↗
Extrapolation techniques applied to matrix methods in neutron diffusion problems
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Mccready, Robert R↗
Application of the Space-Time Conservation Element and Solution Element Method to One-Dimensional Convection-Diffusion Problems
In the space-time conservation element and solution element (CE/SE) method, the independent marching variables used comprise not only the mesh value of the physical dependent variables but also, in contrast to it typical numerical method, the Mesh values of the spatial derivatives of the physical variables The use of the extra marching variables results from the need to construct the two-level explicit and nondissipative schemes which are at the core of the CE/SE development. It also results from the need to minimize the stencil while maintaining accuracy. In this paper using the 1D(sub (alpha)-mu) scheme as an example, the effect of this added complication on consistency, accuracy and operation count is assessed. As part of this effort, an equivalent yet more efficient form of the alpha-mu scheme in which the independent marching variables are the local fluxes tied to each mesh point is introduced. Also, the intriguing relations that exist among the alpha-mu. Leapfrog, and DuFort-Frankel schemes are further explored. In addition, the redundance of the Leapfrog, DUFort-Frankel, and Lax scheme and the remedy for this redundance are discussed. This paper is concluded with the construction and evaluation of a CE/SE solver for the inviscid Burger equation.
Chang, Sin-Chung↗
Randomized problem of diffuse reflection
Randomized problem of diffuse reflection, and radiative transfer in presence of both scattering and absorption
PARTICLE DIFFUSION↗
Solution techniques for incompressible flow problems
A three-step Petrov-Galerkin (PG)/operator spliting scheme for the time-dependent incompressible Navier-Stokes equations is proposed. Each time step is split into two Stokes problems and one nonlinear convection-diffusion problem. Using a PG technique on the two outer Stokes problems ensures a stable scheme despite equal-order interpolation, while using a streamline upwind PG scheme on the inner convection-diffusion problem ensures a numerically stable solution at high Reynolds numbers. Numerical tests of this method have been carried out.
Tezduyar, T. E.↗
A Systematic Solution Approach for Neutron Transport Problems in Diffuse Regimes
A systematic solution approach for the neutron transport equation, based on a least-squares finite-element discretization, is presented. This approach includes the theory for the existence and uniqueness of the analytical as well as of the discrete solution, bounds for the discretization error, and guidance for the development of an efficient multigrid solver for the resulting discrete problem. To guarantee the accuracy of the discrete solution for diffusive regimes, a scaling transformation is applied to the transport operator prior to the discretization. The key result is the proof of the V-ellipticity and continuity of the scaled least-squares bilinear form with constants that are independent of the total cross section and the absorption cross section. For a variety of least-squares finite-element discretizations this leads to error bounds that remain valid in diffusive regimes. Moreover, for problems in slab geometry a full multigrid solver is presented with V(1, 1)-cycle convergence rates approximately equal to 0.1, independent of the size of the total cross section and the absorption cross section.
Manteuffel, T. A.↗
Comments on the Diffusive Behavior of Two Upwind Schemes
The diffusive characteristics of two upwind schemes, multi-dimensional fluctuation splitting and locally one-dimensional finite volume, are compared for scalar advection-diffusion problems. Algorithms for the two schemes are developed for node-based data representation on median-dual meshes associated with unstructured triangulations in two spatial dimensions. Four model equations are considered: linear advection, non-linear advection, diffusion, and advection-diffusion. Modular coding is employed to isolate the effects of the two approaches for upwind flux evaluation, allowing for head-to-head accuracy and efficiency comparisons. Both the stability of compressive limiters and the amount of artificial diffusion generated by the schemes is found to be grid-orientation dependent, with the fluctuation splitting scheme producing less artificial diffusion than the finite volume scheme. Convergence rates are compared for the combined advection-diffusion problem, with a speedup of 2.5 seen for fluctuation splitting versus finite volume when solved on the same mesh. However, accurate solutions to problems with small diffusion coefficients can be achieved on coarser meshes using fluctuation splitting rather than finite volume, so that when comparing convergence rates to reach a given accuracy, fluctuation splitting shows a speedup of 29 over finite volume.
Wood, William A.↗