##### Document Text Contents

Page 1

Computational

Heat

Transfer

Volume 1

MaShematScal Modslling

A.A. SAMARSKll

P.N. VABISHCHEVICH

Page 2

Computational Heat

Transfer

Volume 1 Mathematical Modelling

A. A. Samarskii

P. N. Vabishchevich

Russian Academy of Sciences, Moscow

JOHN WILEY & SONS

Chichester New York Brisbane Toronto Singapore

Page 208

STATIONARY PROBLEMS OF HEAT TRANSFER 197

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.

4.8.4 DECOMPOSITION METHODS WITHOUT OVERLAPPING

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.

Page 209

198 COMPUTATIONAL HEAT TRANSFER

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

INDEX

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

solutions

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

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

temperature

liquidus temperature 34

solidus temperature 34

thermal conductivity 16

thermal diffusivity 17

tbermodiffusion Stefan problem 392

Page 417

406 COMPUTATIONAL HEAT TRANSFER

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

transform

fast Fourier transform 159

Goodman transform 70

integral transform 67

Kirchhoff transform 70

transformation

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

Computational

Heat

Transfer

Volume 1

MaShematScal Modslling

A.A. SAMARSKll

P.N. VABISHCHEVICH

Page 2

Computational Heat

Transfer

Volume 1 Mathematical Modelling

A. A. Samarskii

P. N. Vabishchevich

Russian Academy of Sciences, Moscow

JOHN WILEY & SONS

Chichester New York Brisbane Toronto Singapore

Page 208

STATIONARY PROBLEMS OF HEAT TRANSFER 197

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.

4.8.4 DECOMPOSITION METHODS WITHOUT OVERLAPPING

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.

Page 209

198 COMPUTATIONAL HEAT TRANSFER

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

INDEX

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

solutions

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

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

temperature

liquidus temperature 34

solidus temperature 34

thermal conductivity 16

thermal diffusivity 17

tbermodiffusion Stefan problem 392

Page 417

406 COMPUTATIONAL HEAT TRANSFER

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

transform

fast Fourier transform 159

Goodman transform 70

integral transform 67

Kirchhoff transform 70

transformation

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