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非线性发展方程的有限差分方法(英文版)
  • 书号：9787030747914
    作者：孙志忠,张启峰,高广花
  • 外文书名：
  • 装帧：圆脊精装
    开本：B5
  • 页数：416
    字数：450000
    语种：zh-Hans
  • 出版社：科学出版社
    出版时间：2023-05-01
  • 所属分类：方程/动力系统
  • 定价： ￥168.00元
    售价： ￥109.20元
  • 图书介质:
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本书针对应用科学中的介绍了12个重要的非线性演化方程的有限差分方法的最新研究成果，包括微分方程问题解的守恒性和有界性分析、差分方法的建立、差分解的守恒性和有界性分析、差分解的存在性分析、差分解收敛性的证明、差分格式的求解等内容。建立的差分求解格式包括非线性差分格式和线性化差分格式。12个非线性演化方程如下：Fisher方程、Burgers方程、正则长波方程、Korteweg-deVries方程、Camassa-Holm方程、Schr.dinger方程、Kuramoto-Tsuzuki方程、Zakharov方程、Ginzburg-Landau方程、Cahn-Hilliard方程、外延增长模型方程和相场晶体模型方程。
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目录

  • Contents
    Preface Ill
    About the Authors V
    1 Difference methods for the Fisher equation 1
    1.1 Introduction 1
    1.2 Notation and lemmas 3
    1.3 Forward Euler difference scheme 11
    1.3.1 Derivation of the difference scheme 11
    1.3.2 Solvability and convergence of the difference scheme 12
    1.4 Backward Euler difference scheme 14
    1.4.1 Derivation of the difference scheme 14
    1.4.2 Existence and convergence of the difference solution 16
    1.5 Crank-Nicolson difference scheme 18
    1.5.1 Derivation of the difference scheme 19
    1.5.2 Existence and convergence of the difference solution 20
    1.6 Fourth-order compact difference scheme 22
    1.6.1 Derivation of the difference scheme 22
    1.6.2 Existence and convergence of difference solution 23
    1.7 Three-level linearized difference scheme 27
    1.7.1 Derivation of the difference scheme 27
    1.7.2 Existence and convergence of the difference solution 31
    1.8 Numerical experiments 36
    1.9 Summary and extension 38
    2 Difference methods for the Burgers’ equation 41
    2.1 Introduction 41
    2.2 Two-level nonlinear difference scheme 43
    2.2.1 Derivation of the difference scheme 43
    2.2.2 Conservation and boundedness of the difference solution 44
    2.2.3 Existence and uniqueness of the difference solution 46
    2.2.4 Convergence of the difference solution 49
    2.3 Three-level linearized difference scheme 54
    2.3.1 Derivation of the difference scheme 54
    2.3.2 Existence and uniqueness of the difference solution 55
    2.3.3 Conservation and boundedness of the difference solution 56
    2.3.4 Convergence of the difference solution 57
    2.4 Hopf-Cole transformation and fourth-order difference scheme 61
    2.4.1 Hopf-Cole transformation 61
    2.4.2 Derivation of the difference scheme 63
    2.4.3 Existence and uniqueness of the difference solution 65
    2.4.4 Convergence of the difference solution 67
    2.4.5 Calculation of the solution of the original problem 69
    2.5 Fourth-order compact two-level nonlinear difference scheme 70
    2.5.1 Derivation of the difference scheme 73
    2.5.2 Conservation and boundedness of the difference solution 74
    2.5.3 Existence and uniqueness of the difference solution 75
    2.5.4 Convergence of the difference solution 79
    2.5.5 Stability of the difference solution 84
    2.6 Numerical experiments 87
    2.7 Summary and extension 88
    3 Difference methods for the regularized long-wave equation 91
    3.1 Introduction 91
    3.2 Two-level nonlinear difference scheme 92
    3.2.1 Derivation of the difference scheme 92
    3.2.2 Existence of the difference solution 93
    3.2.3 Conservation and boundedness of the difference solution 94
    3.2.4 Uniqueness of the difference solution 95
    3.2.5 Convergence of the difference solution 96
    3.3 Three-level linearized difference scheme 98
    3.3.1 Derivation of the difference scheme 98
    3.3.2 Conservation and boundedness of the difference solution 99
    3.3.3 Existence and uniqueness of the difference solution 100
    3.3.4 Convergence of the difference solution 100
    3.4 Numerical experiments 103
    3.5 Summary and extension 104
    4 Difference methods for the Korteweg-de Vries equation 106
    4.1 Introduction 106
    4.2 First-order in space two-level nonlinear difference scheme 107
    4.2.1 Derivation of the difference scheme 107
    4.2.2 Existence of the difference solution 110
    4.2.3 Conservation and boundedness of the difference solution 112
    4.2.4 Convergence of the difference solution 113
    4.3 First-order in space three-level linearized difference scheme 115
    4.3.1 Derivation of the difference scheme 115
    4.3.2 Existence and uniqueness of the difference solution 116
    4.3.3 Conservation and boundedness of the difference solution 117
    4.3.4 Convergence of the difference solution 118
    4.4 Second-order in space two-level nonlinear difference scheme 123
    4.4.1 Derivation of the difference scheme 123
    4.4.2 Existence of the difference solution 125
    4.4.3 Conservation and boundedness of the difference solution 127
    4.4.4 Convergence and uniqueness of the difference solution 128
    4.5 Second-order in space three-level linearized difference scheme 137
    4.5.1 Derivation of the difference scheme 137
    4.5.2 Conservation and boundedness of the difference solution 139
    4.5.3 Existence and uniqueness of the difference solution 141
    4.5.4 Convergence of the difference solution 143
    4.6 Numerical experiments 146
    4.7 Summary and extension 148
    5 Difference methods for the Camassa-Holm equation 151
    5.1 Introduction 151
    5.2 Two-level nonlinear difference scheme 152
    5.2.1 Derivation of the difference scheme 152
    5.2.2 Conservation of the difference solution 153
    5.2.3 Existence and uniqueness of the difference solution 154
    5.2.4 Convergence of the difference solution 157
    5.3 Three-level linearized difference scheme 159
    5.3.1 Derivation of the difference scheme 159
    5.3.2 Conservation and boundedness of the difference solution 160
    5.3.3 Existence and uniqueness of the difference solution 161
    5.3.4 Convergence of the difference solution 162
    5.4 Numerical experiments 168
    5.5 Summary and extension 172
    6 Difference methods for the Schrodinger equation 174
    6.1 Introduction 174
    6.2 Two-level nonlinear difference scheme 176
    6.2.1 Derivation of the difference scheme 176
    6.2.2 Conservation and boundedness of the difference solution 177
    6.2.3 Existence and uniqueness of the difference solution 180
    6.2.4 Convergence of the difference solution 182
    6.3 Three-level linearized difference scheme 188
    6.3.1 Derivation of the difference scheme 188
    6.3.2 Conservation and boundedness of the difference solution 189
    6.3.3 Existence and uniqueness of the difference solution 191
    6.3.4 Convergence of the difference solution 192
    6.4 Fourth-order three-level linearized difference scheme 200
    6.4.1 Several numerical differential formulas 200
    6.4.2 Derivation of the difference scheme 203
    6.4.3 Existence and uniqueness of the difference solution 205
    6.4.4 Conservation and boundedness of the difference solution 207
    6.4.5 Convergence of the difference solution 211
    6.5 Numerical experiments 216
    6.6 Summary and extension 219
    7 Difference methods for the Kuramoto-Tsuzuki equation 220
    7.1 Introduction 220
    7.2 Two-level nonlinear difference scheme 225
    7.2.1 Derivation of the difference scheme 225
    7.2.2 Existence of the difference solution 227
    7.2.3 Boundedness of the difference solution 228
    7.2.4 Uniqueness of the difference solution 233
    7.2.5 Convergence of the difference solution 234
    7.3 Three-level linearized difference scheme 237
    7.3.1 Derivation of the difference scheme 237
    7.3.2 Boundedness of the difference solution 239
    7.3.3 Existence and uniqueness of the difference solution 241
    7.3.4 Convergence of the difference solution 242
    7.4 Numerical experiments 246
    7.5 Summary and extension 247
    8 Difference methods for the Zakharov equation 249
    8.1 Introduction 249
    8.2 Two-level nonlinear difference scheme 253
    8.2.1 Derivation of the difference scheme 253
    8.2.2 Existence of the difference solution 255
    8.2.3 Conservation and boundedness of the difference solution 257
    8.2.4 Convergence of the difference solution 259
    8.3 Three-level linearized locally decoupled difference scheme 267
    8.3.1 Derivation of the difference scheme 267
    8.3.2 Existence of the difference solution 269
    8.3.3 Conservation and boundedness of the difference solution 270
    8.3.4 Convergence of the difference solution 274
    8.4 Numerical experiments 282
    8.5 Summary and extension 283
    9 Difference methods for the Ginzburg-Landau equation 285
    9.1 Introduction 285
    9.2 Two-level nonlinear difference scheme 286
    9.2.1 Derivation of the difference scheme 290
    9.2.2 Existence of the difference solution 291
    9.2.3 Boundedness of the difference solution 292
    9.2.4 Convergence of the difference solution 294
    9.3 Three-level linearized difference scheme 298
    9.3.1 Derivation of the difference scheme 298
    9.3.2 Existence of the difference solution 299
    9.3.3 Boundedness of the difference solution 300
    9.3.4 Convergence of the difference solution 302
    9.4 Numerical experiments 307
    9.5 Summary and extension 308
    10 Difference methods for the Cahn-Hilliard equation 309
    10.1 Introduction 309
    10.2 Two-level nonlinear difference scheme 312
    10.2.1 Derivation of the difference scheme 315
    10.2.2 Existence of the difference solution 317
    10.2.3 Boundedness of the difference solution 319
    10.2.4 Convergence of the difference solution 320
    10.3 Three-level linearized difference scheme 326
    10.3.1 Derivation of the difference scheme 326
    10.3.2 Existence and uniqueness of the difference solution 327
    10.3.3 Convergence of the difference solution 328
    10.4 Three-level linearized compact difference scheme 336
    10.4.1 Derivation of the difference scheme 338
    10.4.2 Existence and uniqueness of the difference solution 340
    10.4.3 Convergence of the difference solution 342
    10.5 Numerical experiments 348
    10.6 Summary and extension 349
    11 Difference methods for the epitaxial growth model 351
    11.1 Introduction 351
    11.2 Notation and basic lemmas 352
    11.3 Two-level nonlinear backward Euler difference scheme 356
    11.3.1 Derivation of the difference scheme 356
    11.3.2 Boundedness of the difference solution 357
    11.3.3 Existence and uniqueness of the difference solution 359
    11.3.4 Convergence of the difference solution 362
    11.4 Two-level linearized backward Euler difference scheme 365
    11.4.1 Derivation of the difference scheme 366
    11.4.2 Boundedness of the difference solution 367
    11.4.3 Existence of the difference solution 367
    11.4.4 Convergence of the difference solution 368
    11.5 Three-level linearized backward Euler difference scheme 371
    11.5.1 Derivation of the difference scheme 371
    11.5.2 Boundedness of the difference solution 373
    11.5.3 Existence of the difference solution 376
    11.5.4 Convergence of the difference solution 377
    11.6 Numerical experiments 382
    11.7 Summary and extension 385
    12 Difference methods for the phase field crystal model 387
    12.1 Introduction 387
    12.2 Notation and basic lemmas 388
    12.3 Two-level nonlinear difference scheme 390
    12.3.1 Derivation of the difference scheme 390
    12.3.2 Boundedness of the difference solution 391
    12.3.3 Existence and uniqueness of the difference solution 393
    12.3.4 Convergence of the difference solution 396
    12.4 Three-level linearized difference scheme 399
    12.4.1 Derivation of the difference scheme 399
    12.4.2 Energy stability of the difference solution 400
    12.4.3 Convergence of the difference solution 402
    12.5 Numerical experiments 406
    12.6 Summary and extension 407
    Bibliography 411
    Index 415
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