The Theory of Difference Schemes - Alexander A. Samarskii

The Theory of Difference Schemes......Page 1
Preface......Page 9
Contents......Page 20
1. Preliminary comments......Page 23
2. Examples of difference equations......Page 24
3. The first-order difference equations and inequalities......Page 27
4.The second-order difference equations.The Cauchy problem.Bounda-\rry-value problems.......Page 29
5.The elimination method......Page 31
6.Stability of the elimination method.......Page 33
7.The left elimination method.The counter elimination method.......Page 35
8.Maximum principle.......Page 36
9.Maximum principle for the third kind boundary-value problem.......Page 42
10.Solution estimation for difference boundary-value problems by the elim-ination method.......Page 43
11.The second-order difference equations with constant coefficients.......Page 46
12.Formulae of \"difference differentiation\"by parts of the product and sum.......Page 51
13.Green\'s difference formulae.......Page 53
1.The flow variant of the elimination method for difference problems with widely varying coefficients.......Page 56
2.Cyclic elimination method......Page 59
3.Factorization method.......Page 61
2.1. RELEVANT ELEMENTS OF FUNCTIONAL ANALYSIS......Page 63
2.2 DIFFERENCE APPROXIMATION OF ELEMENTARY\rDIFFERENTIAL OPERATORS......Page 72
1. Examples of Stable and Unstable Difference Schemes.......Page 109
2.The Cauchy problem for a system of differential equations of first order.......Page 112
4.Stability, approximation and convergence.......Page 118
1.Some difference formulae.......Page 120
2.The search for eigenfunctions and eigenvalues in the example of the simplest difference problem.......Page 124
3.Eigenvalue problems with boundary conditions of the second and third kinds.......Page 128
4.Difference analogs of the embedding theorems.......Page 132
5. The method of energy inequalities.......Page 135
2.5 DIFFERENCE SCHEMES AS OPERATOR EQUATIONS. GENERAL FORMULATIONS......Page 138
1.Difference schemes as operator equations.......Page 139
2.Stability of a difference scheme.......Page 148
3.Convergence and approximation.......Page 150
4.Some a priori estimates.......Page 154
5.Negative norms.......Page 158
6.Operator equations of divergent type.......Page 162
1. Introduction......Page 167
1.An example of the scheme which is divergent in the case of discontinuous coefficients.......Page 169
2.The integro-interpolational method for constructing homogeneous dif- ference schemes.......Page 172
3.Homogeneous conservative schemes.......Page 176
4.A primary family of conservative schemes.......Page 178
1.The error of approximation in the class of smooth coefficients.......Page 181
2.The error of approximation in the class of discontinuous coefficients.......Page 182
3.A priori estimates of the error......Page 186
4.On convergence and accuracy.......Page 188
1.Schemes on non-equidistant grids.......Page 190
2.The error of approximation......Page 192
3.The order of accuracy on non-equidistant grids.......Page 194
4.A greater gain in accuracy on a sequence of grids.The Runge method.......Page 196
1.The third boundary-value problem......Page 200
2.A problem with periodicity conditions......Page 201
3.Monotone schemes for an equation of general form.......Page 205
4.Difference schemes for a stationary equation in cylindrical coordinates.......Page 209
5.Difference schemes for an equation in spherical coordinates......Page 218
1.Difference Green\'s function.......Page 221
2.A priori estimates.......Page 225
1.An exact scheme.......Page 229
2.Schemes of arbitrary order accuracy.......Page 234
1.General remarks.......Page 236
2.The integro-interpolational method (HM)......Page 237
3.Variational difference methods (the Ritz method and the Bubnov Galerkin method)......Page 243
4.The method of approximating a quadratic functional......Page 247
5.The method of the summator identities (the method of approximating an integral identity)......Page 250
1.Stability of difference schemes with respect to coefficients.......Page 251
2.Stability of the first kind operator equations with respect to coefficients.......Page 254
4.1 THE DIRICHLET DIFFERENCE PROBLEM FOR POISSON\'S EQUATION......Page 258
1.The difference approximation of the Laplace operator.......Page 259
2.Approximation of the Laplace operator on an irregular \"cross\"pattern.......Page 261
3.The Dirichlet difference problem in a rectangle.......Page 266
4.The Dirichlet difference problem in a domain of rather complicated configuration.......Page 269
5.The canonical form of a difference equation.......Page 276
1.The canonical form of a grid equation of common structure.......Page 279
2.The maximuin principle.......Page 281
3.Comparison theorem.The majoraiit.......Page 283
4.The estimate of a solution to the nonhomogeneous equation.......Page 284
1.Estimation of a solution of the Dirichlet difference problem.......Page 286
2.The uniform convergence and the order of accuracy of a difference scheme.......Page 292
1.Eigenvalue problems for the Laplace difference operator in a rectan- gle.......Page 293
2.Properties of difference operators.......Page 298
3.The Laplace operator in a domain consisting of rectangles.......Page 301
4.The embedding theorem.......Page 302
5.Equations with variable coefficients.......Page 304
6.Equations with mixed derivatives.......Page 307
1.The statement of the Dirichlet difference problem providing a higher- order approximation.......Page 311
2.Estimation of a solution of the difference boundary-value problem.......Page 315
3.The multidimensional case.......Page 318
5.1 ONE-DIMENSIONAL HEAT CONDUCTION EQUATION WITH CONSTANT COEFFICIENTS......Page 320
1.The original problem.......Page 321
2.The family of six-point schemes.......Page 322
3.The error of approximation.......Page 324
4.Stability with respect to the initial data.......Page 327
5.Stability with respect to the right-hand side.......Page 332
6.Convergence and accuracy in the space L.2 (omega h)......Page 334
7.Stability and convergence in the space C.......Page 335
8.The method of energy inequalities.......Page 337
9.The third kind boundary conditions.......Page 342
10.Three-layer schemes for the heat conduction equation.......Page 344
1.Rho-stability.......Page 348
2.Asymptotic stability.......Page 351
3.The scheme of second-order accuracy (unconditionally stable in the as- ymptotic sense).......Page 355
4.Asymptotic stability of the three-layer scheme.......Page 359
1.The explicit difference scheme.......Page 361
2.The explicit three-layer scheme.......Page 366
3.Schemes with weights.......Page 367
4.The scheme with increased accuracy.......Page 368
1.Two-layer scheme with weights......Page 370
2.Three-layer schemes......Page 374
1.Explicit schemes for the Cauchy problem.......Page 375
2.Explicit schemes of a higher-order approximation......Page 379
3.The boundary-value problem......Page 381
1.The statement of the difference problem and calculations of the approx- imation error.......Page 385
2.Stability analysis.......Page 388
3.The energy inequality method.......Page 395
4.Determination of nonsmooth solutions by the difference method.......Page 397
5.7 SELECTED PROBLEMS......Page 399
1.Introduction.......Page 403
2.Operator-difference schemes.......Page 404
3.The canonical form of two-layer schemes.......Page 405
4.The canonical form of three-layer schemes.......Page 407
5.The notion of stability.......Page 408
6.Sufficient stability conditions for two-layer schemes in linear normed spaces.......Page 410
7.Approximation and convergence.......Page 414
1.The problem statement.......Page 417
2.A primary family of schemes.......Page 418
4.Stability with respect to the initial data in HA......Page 419
5.Stability with respect to the initial data in HB.......Page 423
6.Estimates of the norm of the transition operator.......Page 424
7.The method of separation of variables.......Page 426
8.The rho-stability condition.......Page 428
9.Stability with respect to the right-hand side.......Page 431
10.Stability of the weighted scheme.......Page 436
11.A priori estimates in the case of a variable operator A.......Page 440
12.Example.......Page 443
13.The case of a skew-symmetric operator A.......Page 445
1.The problem statement.......Page 448
2.The basic energy identity.......Page 450
3.Stability with respect to the initial data.......Page 452
4.Stability with respect to the right-hand side.......Page 455
5.Schemes with variable operators.......Page 458
6.Weighted schemes.......Page 461
7.Examples.......Page 464
8.Three-layer schemes with non-self-adjoint operators.......Page 465
9.Other a priori estimates.......Page 468
10.On regularization of difference schemes.......Page 474
1.The original problem.......Page 479
2.Homogeneous difference schemes with weights......Page 480
3.Stability and convergence.......Page 484
4.The equation with a discontinuous coefficient of heat conductivity......Page 489
5.Homogeneous schemes on nonequidistant grids.......Page 495
6.An one-point heat source.......Page 501
7.A concentrated heat capacity.......Page 503
8.The case when the coefficient of conductivity k depends on t,k =k(x,t}......Page 507
9.The third boundary-value problem.......Page 509
10.Monotone schemes for parabolic equations of general form.......Page 510
11.Cylindrically symmetric and spherically symmetric heat conduction problems.......Page 513
12.A periodic problem.......Page 516
2.Homogeneous difference schemes.......Page 519
3.The error of approximation.......Page 521
4.Stability and convergence.......Page 523
1.The stationary problem.......Page 527
2.The quasilinear heat conduction equation.......Page 531
3.Some analytical solutions to the quasiliiiear heat conduction equation.......Page 532
4.A difference scheme.Newton\'s method.......Page 537
5.Various implicit schemes for the quasilinear heat conduction equation.......Page 540
7.The Stephan problem (problem of the phase transition)......Page 543
1.One-dimensional equations of nonstationary gas dynamics in Lagraiigiaii variables......Page 545
2.Equations with psevdoviscosity......Page 548
3.Conservative homogeneous schemes......Page 549
4.Fully conservative schemes......Page 552
5.Numerical solution of difference equations by Newton\'s method......Page 556
6.Convergence of Newton\'s method.......Page 557
7.Equations of gas dynamics with heat conductivity......Page 561
1.Some preliminary information on economical schemes......Page 563
2.An alternating direction scheme......Page 567
3.Stability......Page 570
4.Stability and accuracy......Page 573
5.A scheme for the governing equation with variable coefficients......Page 574
6.A higher-accuracy scheme......Page 580
1.Schemes with factorized operator......Page 584
2.The boundary conditions.......Page 587
3.Constructions of economical factorized schemes......Page 588
4.Three-layer factorized schemes......Page 593
5.Economical schemes for a system of equations of parabolic and hyperbolic types......Page 602
1.The problem statement......Page 611
2.The notion of summarized approximation......Page 612
3.Reduction of a multidimensional problem to a chain of one-dimensional problems......Page 616
4.Examples of reduction of multidimensional problems to chains of one- dimensional ones......Page 620
5.A locally one-dimensional scheme (LOS)for the heat conduction equation in an arbitrary domain......Page 624
6.The error of approximation of a locally one-dimensional scheme......Page 628
7.Stability of LOS......Page 630
8.Uniform convergence of LOS......Page 634
9.LOS for equations with variable coefficients......Page 636
10.Additive schemes.The general formulations and statements......Page 637
11.Methods for the convergence rates of additive schemes......Page 640
12.An approximation of the \"multidimensional\"abstract Cauchy problem by a chain of the \"one-dimensional\"Cauchy problems......Page 643
13.Alternating direction methods as additive schemes......Page 649
14.LOS for a multidimensional hyperbolic equation of second order......Page 650
15.Additive schemes for a system of equations......Page 656
1.Direct and iterative methods......Page 663
2.The decomposition method......Page 665
3.The method of separation of variables......Page 670
4.The method of matrix elimination......Page 671
1.Two-layer iteration schemes.The problem statement......Page 673
2.A stationary scheme.The main theorem on the convergence of itera- tions......Page 677
3.The explicit scheme with optimal set of Chebyshev\'s parameters......Page 680
4.The simple iteration scheme......Page 684
5.A model problem.Comparison of methods.......Page 685
6.On computational stability of iterative methods......Page 687
7.Re-ordering of iteration parameters......Page 691
1.Seidel method.......Page 696
2.The upper relaxation method......Page 698
3.Implicit iteration schemes......Page 699
4.The alternative triangle method......Page 703
5.ATM for the difference Dirichlet problem......Page 707
6.A higher-accuracy scheme in a rectangle......Page 712
7.Difference schemes for elliptic equations of general form......Page 714
8. ATM for solving grid elliptic equations in an arbitrary complex domain......Page 719
9.The Dirichlet problem for Poisson\'s equation in an arbitrary complex do- main......Page 726
10.On solving difference equations for problems with variable coefficients......Page 728
1.The alternating direction method for solving the difference Dirichlet problem in a rectangle......Page 730
2.The general framework of ADM.......Page 732
3.A proper choice of iteration parameters by Jordan\'s rule......Page 734
4.ADM for the case of noncommutative operators......Page 737
5.Factorized iteration schemes and ADM......Page 742
6.ADM for non-self-adjoint operators......Page 746
1.Three-layer iteration schemes......Page 749
2.The minimal residual method......Page 752
3.The method of steepest descent......Page 754
4.On solving equations with iion-self-adjoint operators......Page 756
5.Gibrid (combined)methods......Page 761
Symbols......Page 765
Concluding Remarks......Page 768
References......Page 771