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Page 1


Volume 1

MaShematScal Modslling


Page 2

Computational Heat
Volume 1 Mathematical Modelling

A. A. Samarskii

P. N. Vabishchevich

Russian Academy of Sciences, Moscow

Chichester New York Brisbane Toronto Singapore

Page 208


Large difference in coefficients of an elliptic operator is typical of boundary
value problems like (16), (17) solved by the method of fictitious domains.
In the version with extension witb respect to the principal coefficients (see
e.g. (18)) a large (small) parameter multiplies the leading derivatives, and we
thus encounter a singularly perturbed problem. The version with extension
with respect to the lowest coefficients is not associated with such essential
transformation of the problem. Hence, the latter is preferable for applied
mathematical modelling.

When constructing difference schemes of the method of fictitious domains,
it is useful to apply the integreinterpolation method, which was developed
just for problems with discontinuous coefficients. Grid problems are solved by
various iterative methods. As for the convergence rate of iterative processes,
the versions with extension witb respect to lowest coefficients are preferable
as compared with those employing extension with respect to principal terms.


A decomposition method, in which the computational domain is separated
into simple subdomains, gives another approach to solving boundary value
problems in irregular conlputational domains. This approach is actively
discussed as applied to developing methods for solving boundary value
processes on parallel computers.

In each subdomain, its own problem is solved. The domains are connected
via boundary conditions. In each subdomain, its own grid is introduced, which
can be inconsistent with grids in other subdomains. Therefore, difference
schemes of decomposition methods can be treated as difference methods on
composite grids.

There are two important classes of decomposition methods. The first is
associated with partitioning the domain into nonoverlapping subdomains. In
the second class of methods, separate subdomains may intersect one another.

Fig. 4.9a. Fig. 4.9b.

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In order to explain the essence of the decomposition methods, we consider
a model problem in an L-shaped (Fig. 4.9) domain. It is convenient to divide
this irregular domain R into two regular subdomains (rectangles) nl and
0 2 without overlapping. Figure 4.9 shows two simple versions of such a
partitioning of the L-shaped domain. In order to solve difference problems in
the subdomains, in several cases we can employ fast direct methods for solving
grid equations considered in Section 4.5. Therefore, particular attention in the
following is paid to methods for determining the approximate solution on the
common boundary y of the subdomains R1 and Rz (y = awl n 802).

Assume that boundary value problem (lo), (11) is solved in the domain S1.
As for the method of fictitious domains, we only consider the main points of
the decomposition method from the viewpoint of differential calculus. If the
boundary condition on y were known, the solution of the problem in R would
be reduced to solving two separate problems in subdomains 01 and Rz. Let
us denote the exact solution of problem (lo), (11) in the subdomain Rl by
ul(x) and that in R2 by u ~ ( x ) . The junction condition on the boundary y in
this notation becomes

Here n,, a = 1,2, is the external normal relative t o the domain S1,, u = 1,2.
Let us construct the simplest iterative process to explicitly refine the first-kind
boundary condition on y taking into account (30) by the formula

where u(s) = ul(z) = u2(x). The boundary condition is refined according to
the flux unbalance. In each separate subdomain R,, u = 1,2, we solve the
boundary value problem

= s(x), 2: E an,\y, (32)

u = u x , X E y.

An alternative to (31) is iterative refinement of the second-kind boundary
condition (using the temperature unbalance), namely

Page 416


searching experiment 11
second difference Green formula 140
second-kind boundary conditions 23
Seidel method 177
self-adjoint operator 135
self-similar solutions 72
singularly perturbed problem 78
skew-symmetric operator 135
solidus temperature 34

approximation of the boundary condition on the solutions 105
approximation on the solutions 119
self-similar solutions 72

sparse matrix 153
specific heat capacity 16
spectral problem 63
stability of the difference scheme 100
stability of the difference scheme with respect t o the initial data 233
stability of the difference scheme with respect t o the right-hand side 234
stabilization-correction difference scheme 313
stable difference scheme 233
steady-state iterative method 164
Stefan number 62
Stefan problem 32
Stefan-Boltzmann law 46
Steklov averaging operators 113
stencil 101
stream function 39

stability of the difference scheme 100
stability of the difference scheme with respect to the initial data 233
stability of the difference scheme with respect to the right-hand side 234
uniform stability with respect to the initial data 234

summarized approximation 327
symmetrical difference scheme (the Crank-Nicolson scheme) 241
symmetrical over-relaxation method 180

liquidus temperature 34
solidus temperature 34

thermal conductivity 16
thermal diffusivity 17
tbermodiffusion Stefan problem 392

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third-kind boundary condition 24
Thomas algorithm 155
three-layer iterative method 162
three-level difference scheme 231
three-level difference scheme with weights 245

fast Fourier transform 159
Goodman transform 70
integral transform 67
Kirchhoff transform 70

D u ~ u t transformation 371

triangular iterative method 177
two-level difference scheme 231
two-level iterative method 162
two-phase Stefan problem 32
two-step (three-level) iterative method 162
two-step iterative method 213

uniform stability with respect to the initial data 234

variable domain method 347
variable inversion method 385
variational iterative method 166
viscosity 38

well-posed problem 27

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