Singular Source Terms Research Articles

On the approximation of singular source terms in differential equations

We study differential equations with singular source terms. For such equations classical convergence results do not apply, as these rely on the regularity of the solution and the source terms. We study some elliptic and parabolic problems numerically and theoretically, and show that, with the right approximation of the singular source terms, full convergence order can be achieved away from the singularities, whereas the convergence will be poor in a vicinity of these. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 503–520, 1999

On the existence of shock fronts for a Burgers equation with a singular source term

In this paper we consider the Cauchy problem for the equation ∂u/∂t + u ∂u/∂x + u/x = 0 for x > 0, t ⩾ 0, with u(x, 0) = u0−(x) for x < x0, u(x, 0) = u0+(x) for x > x0, u0−(x0) > u0+(x0). Following the ideas of Majda, 1984 and Lax, 1973, we construct, for smooth u0− and u0+, a global shock front weak solution u(x, t) = u−(x, t) for x < ϕ(t), u(x, t) = u+(x, t) for x > ϕ(t), where u− and u+ are the strong solutions corresponding (respectively) to u0− and u0+ and the curve t → ϕ(t) is defined by dϕ/dt (t) = 1/2[u−(ϕ(t), t) + u+(ϕ(t), t)], t ⩾ 0 and ϕ(0) = x0. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.

Determination of temperature field around a rapidly moving crack-tip in an elastic-plastic solid

The problem of local heating and temperature rise induced by dynamic crack growth in elastic-plastic solids is studied numerically. Heat generation caused by plastic work dissipation is estimated from crack-tip stress and deformation fields obtained separately by two of the authors. The temperature field in an Eulerian description is shown to be governed by a convection-dominated flow equation with a singular source term that is distributed over an irregular crack-tip region, the active plastic zone. The peak value and spatial distribution of the temperature increase are determined using two independent computer codes, which are developed by the authors based on an integral representation of the temperature field and on an upwind finite element formulation. The accuracy and reliability of the numerical methods and their solutions are studied carefully against exact, closed-form solutions for several specially designed boundary value problems. These methods are used to simulate dynamic fracture tests on AISI 4340 steel specimens, and the predicted temperature contours and maximum values are found to be in good agreement with those measured and estimated experimentally.

On the Fast Fourier Transform of Functions with Singularities

We consider a simple approach for the fast evaluation of the Fourier transform of functions with singularities based on projecting such functions on a subspace of Multiresolution Analysis. We obtain an explicit approximation of the Fourier Transform of generalized functions and develop a fast algorithm based on its evaluation. In particular, we construct an algorithm for the Unequally Spaced Fast Fourier Transform and test its performance in one and two dimensions. The number of operations required by algorithms of this paper is O(N · logN + Np · (− log ϵ)) in one dimension and O(N2 · logN + Np · (− log ϵ)2) in two dimensions, where ϵ is the precision of computation, N is the number of computed frequencies and Np is the number of nodes. We also address the problem of using approximations of generalized functions for solving partial differential equation with singular coefficients or source terms.

Numerical Approximations for a Miscible Displacement Process in Porous Media

A reservoir simulation problem describing a miscible displacement process is stated and analyzed, and a semidiscrete numerical scheme based on a Galerkin method with mesh refinements is presented for its solution. The underlying physical problem involves two miscible, incompressible fluids of equal viscosity flowing in a porous medium. Injection and production wells are rigorously modelled, which leads to a formulation containing nearly singular source and sink terms. The singularities arising in the solution are described in some detail, and it is shown how knowledge of their forms motivate the choice of an appropriate mesh refinement for the numerical scheme. Optimal order quadratic convergence is shown for the resulting semidiscrete approximation using piecewise linear elements.

Microscopic foundation for the hydrodynamic description of flow near a macroparticle

The two-fluid mean-field theory of the preceding paper is applied to the special case of a macroscopic test particle immersed in a quasicontinuous background fluid. Here, as in the preceding investigation, the particle interactions are taken to be pair-additive sums of long-ranged continuous and short-ranged, rigid-core contributions. The kinetic equation for the conditional singlet distribution function of untagged solvent particles is recast in terms of a related smoothly varying function which satisfies a boundary condition at the interface where this fluid comes into contact with the test particle. This formulation of the theory is then used to examine the case when the test particle is much larger and more massive than a solvent particle and when the free path length of the background medium is significantly smaller than the test particle diameter. In this limit the moments of the background fluid kinetic equation reduce to the linearized Navier–Stokes equations for pure solvent and the velocity moment of the boundary condition becomes identical with the macroscopic slip-flow condition of hydrodynamics. From these, one can obtain the usual Stokes’ formula for the frictional force on the test particle. The theory also can be described by a kinetic equation for the background fluid which incorporates singular source terms arising from the rigid-core interactions between the solvent and the test particle. In the case of a large, very massive test particle the moments of this equation are found to be generalizations to high density of the ‘‘source hydrodynamic equations’’ of Cukier, Kapral, Lebenhaft, and Mehaffey.

Quantized singularities in the gravitational field

The Cabibbo-Ferrari derivation of the Dirac charge quantization condition in electromagnetism is extended to the gravitational field. This is accomplished by use of the pathdependent formalism pioneered by Mandelstam. As a result, we find that the Bianchi identity generalizes to include a quantized, singular source term for the dual Riemann tensor. Under reasonable assumptions, this source term is proportional to the divergence of the energy-momentum tensor, leading to a quantized violation of local energy conservation. Specifically, it is found that the magnitude of the time rate of appearance of three-momentum in any volume of three-space must be an integer multiple of 3c4/2G. Some physical aspects of this energy nonconservation are briefly considered.