646 research outputs found
Singularly perturbed hyperbolic problems on metric graphs: asymptotics of solutions
We are interested in evolution phenomena on star-like networks composed of
several branches which vary considerably in physical properties. The initial
boundary value problem for singularly perturbed hyperbolic differential
equation on a metric graph is studied. The hyperbolic equation becomes
degenerate on a part of the graph as a small parameter goes to zero. In
addition, the rates of degeneration may differ in different edges of the graph.
Using the boundary layer method the complete asymptotic expansions of solutions
are constructed and justified.Comment: 18 pages, 3 figure
Numerical approximation of solution derivatives of singularly peprturbed parabolic problems of convection-difffusion type
Numerical approximations to the solution of a linear singularly perturbed parabolic convection-diffusion problem are generated using a backward Euler method in time and an upwinded finite difference operator in space on a piecewise-uniform Shishkin mesh. A proof is given to show first order convergence of these numerical approximations in an appropriately weighted C^1$-norm. Numerical results are given to illustrate the theoretical error bounds
Parameter-uniform numerical method for global solution and global normalized flux of singularly perturbed boundary value problems using grid equidistribution
AbstractIn this paper, we present the analysis of an upwind scheme for obtaining the global solution and the normalized flux for a convection–diffusion two-point boundary value problem. The solution of the upwind scheme is obtained on a suitable nonuniform mesh which is formed by equidistributing the arc-length monitor function. It is shown that the discrete solution obtained by the upwind scheme and the global solution obtained via interpolation converges uniformly with respect to the perturbation parameter. In addition, we prove the uniform first-order convergence of the weighted derivative of the numerical solution on this nonuniform mesh and the uniform convergence of the global normalized flux on the whole domain. Numerical results are presented that demonstrate the sharpness of our results
Green's kernels for transmission problems in bodies with small inclusions
The uniform asymptotic approximation of Green's kernel for the transmission
problem of antiplane shear is obtained for domains with small inclusions. The
remainder estimates are provided. Numerical simulations are presented to
illustrate the effectiveness of the approach.Comment: 39 pages, 6 figure
Asymptotic analysis of solutions to transmission problems in solids with many inclusions
We construct an asymptotic approximation to the solution of a transmission
problem for a body containing a region occupied by many small inclusions. The
cluster of inclusions is characterised by two small parameters that determine
the nominal diameter of individual inclusions and their separation within the
cluster. These small parameters can be comparable to each other. Remainder
estimates of the asymptotic approximation are rigorously justified. Numerical
illustrations demonstrate the efficiency of the asymptotic approach when
compared with benchmark finite element algorithms.Comment: 30 pages, 5 figure
- …