Numerical solution of partial differential equations finite difference methods
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| Format: | Buch |
| Sprache: | English |
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Oxford
Clarendon Press
1985
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| Ausgabe: | Third edition |
| Schriftenreihe: | Oxford applied mathematics and computing science series
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| 100 | 1 | |a Smith, Gordon D. |0 (DE-588)1158286333 |4 aut | |
| 245 | 1 | 0 | |a Numerical solution of partial differential equations |b finite difference methods |c G. D. Smith, Brunel University |
| 250 | |a Third edition | ||
| 264 | 1 | |a Oxford |b Clarendon Press |c 1985 | |
| 300 | |a xi, 337 Seiten |b Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 0 | |a Oxford applied mathematics and computing science series | |
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
| 650 | 7 | |a Equations aux dérivées partielles |2 ram | |
| 650 | 4 | |a Difference equations |x Numerical solutions | |
| 650 | 4 | |a Differential equations, Partial |x Numerical solutions | |
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| adam_text | Contents
NOTATION xiii
1. INTRODUCTION AND FINITE DIFFERENCE
FORMULAE
Descriptive treatment of elliptic equations 1
Descriptive treatment of parabolic and hyperbolic equations 4
Finite difference approximations to derivatives 6
Notation for functions of several variables 8
2. PARABOLIC EQUATIONS: FINITE
DIFFERENCE METHODS, CONVERGENCE,
AND STABILITY
Transformation to non dimensional form 11
An explicit finite difference approximation to dU/dt = d2U/dxz 12
A worked example covering three cases and including com¬
parison tables 13
Crank Nicolson implicit method 19
Worked example including a comparison table 21
Solution of the implicit equations by Gauss s elimination
method 24
The stability of the elimination method 27
A weighted average approximation 28
Derivative boundary conditions 29
Worked examples including comparison tables:
(i) Explicit formula and central differenced boundary con¬
dition 31
(ii) Explicit formula and forward differenced boundary
condition 33
(iii) Implicit formula and central differenced boundary con¬
dition 36
The local truncation error and a worked example 38
Consistency and a worked example illustrating both consis¬
tency and inconsistency 40
Convergence, descriptive treatment and the analysis of an
explicit approximation 43
Stability, descriptive treatment 47
viii Contents
Vector and matrix norms, subordinate matrix norms, p(A)=£
I|A|| 49
A necessary and sufficient condition for stability, ||A||=£l, and
two worked examples 51
Matrix method of analysis, fixed mesh size. 57
A note on the eigenvalues of /(A) and t/i(A)]~1/2(A) 58
The eigenvalues of a common tridiagonal matrix 59
Theorems on bounds for eigenvalues and an application.
(Gerschgorin s theorems) 60
Gerschgorin s circle theorem and the norm of matrix A 62
Stability criteria for derivative boundary conditions using (i)
the circle theorem (ii) ||A|U«1 63
Stability condition allowing exponential growth 66
Stability, von Neumann s method, and three worked examples 67
The global rounding error 71
Lax s equivalence theorem (statement only) and a detailed
analysis of a simple case 72
Finite difference approximations to dU/dt = V2U in cylindrical
and spherical polar co ordinates 75
A worked example involving lim(dU/dx)Ix 77
Exercises and solutions * 79
3. PARABOLIC EQUATIONS: ALTERNATIVE
DERIVATION OF DIFFERENCE EQUATIONS
AND MISCELLANEOUS TOPICS
Reduction to a system of ordinary differential equations 111
A note on the solution of dV/df = AV+b 113
Finite difference approximations via the ordinary differential
equations 115
The Pade approximants to exp 0 116
Standard finite difference equations via the Pade approximants 117
Ao stability, L0 stability and the symbol of the method 119
A necessary constraint on the time step for Crank Nicolson
method 122
The local truncation errors associated with the Pade approxim¬
ants 124
Stiff equations 126
An extrapolation method for improving accuracy in t 126
The symbol for the extrapolation method 128
The arithmetic of the extrapolation method 129 i
The local truncation errors and symbols of extrapolation
schemes 132
Contents ix
The eigenvalue eigenvector solution of a system of ordinary
differential equations
(i) Preliminary results 132
(ii) The eigenvalue eigenvector solution of dV/df = AV 134
(iii) An application giving an approximate solution for
large t 135
Miscellaneous methods for improving accuracy:
(i) Reduction of the local truncation error the Douglas
equations 137
(ii) Use of three time level difference equations 138
(iii) Deferred correction method 139
(iv) Richardson s deferred approach to the limit 141
Solution of non linear parabolic equations:
(i) Newton s linearization method and a worked example 142
(ii) Richtmyer s linearization method 144
(iii) Lee s three time level method 146
A comparison of results for methods (i), (ii), and (iii) for a
particular problem 147
The stability of three or more time level difference equations:
(i) A useful theorem on eigenvalues 148
(ii) Matrices with common eigenvector systems 150
(iii) A worked example 150
Introduction to the analytical solution of homogeneous differ¬
ence equations: 153
(i) The eigenvalues and vectors of a common tridiagonal
matrix 154
(ii) The analytical solution of the classical explicit approxi¬
mation to dU/dt = d2Uldx2 156
Exercises and solutions 158
4. HYPERBOLIC EQUATIONS AND
CHARACTERISTICS
Analytical solution of first order quasi linear equations 175
A worked example and discussion 176
Numerical integration along a characteristic 178
A worked example 179
Finite difference methods on a rectangular mesh for first
order equations:
(i) Lax Wendroff explicit method and a worked example
with a comparison table 181
(ii) Lax Wendroff method for a set of simultaneous equa¬
tions 183
(iii) The Courant Friedrichs Lewy condition 186
x Contents
(iv) Wendrofl s implicit approximation 187
Propagation of discontinuities, first order equations:
(i) discontinuous initial values 188
(ii) Discontinuous initial derivatives 189
Discontinuities and finite difierence approximations. An ex¬
ample using Wendroff s implicit approximation 190
Reduction of a first order equation to a system of ordinary
differential equations 193
The (1,0) Pade difference approximation 195
A comment on the non stiffness of the equations 196
The (1,1) Pade or Crank Nicolson difference equations 196
An improved approximation to dU/dx and the (1,0) Pade
difference equations 197
A word of caution on the central difference approximation to
dU/dx 200
Second order quasi linear hyperbolic equations. Characteristic
curves, and the differential relationship along them 202
Numerical solution by the method of characteristics 204
A worked example 207
A characteristic as an initial curve 209
Propagation of discontinuities, second order equations 210
Finite difference methods on a rectangular mesh for second
order equations: 213
(i) Explicit methods and the Courant Friedrichs Lewy
condition 213
(ii) Implicit methods with particular reference to the wave
equation 216
Simultaneous first order equations and stability 217
Exercises and solutions 220
5. ELLIPTIC EQUATIONS AND SYSTEMATIC
ITERATIVE METHODS
Introduction 239
Worked examples: (i) A torsion problem, (ii) A heat
conduction problem with derivative boundary conditions 240
Finite differences in polar co ordinates 245 .
Improvement of the accuracy of solutions: (i) Finer mesh, (ii)
Deferred approach to the limit, (iii) Deferred correction
method, (iv) More accurate finite difference formulae in ¦,
eluding the nine point formula 248
Analysis of the discretization error of the five point approxi¬
mation to Poisson s equation over a rectangle. Quoted 252
result for irregular boundaries 254
i
Contents xi
Comments on the solution of difference equations, covering
Gauss elimination, LU decomposition, rounding errors, ill
conditioning, iterative refinement, iterative methods 257
Systematic iterative methods for large linear systems 260
Jacobi, Gauss Seidel, and SOR methods 261
A worked example covering each method 263
Jacobi, Gauss Seidel, and SOR methods in matrix form 266
A necessary and sufficient condition for convergence of itera¬
tive methods 268
A sufficient condition for convergence 269
Asymptotic and average rates of convergence 270
Methods for accelerating convergence, (i) Lyusternik s
method, (ii) Aitken s method. An illustrative example 272
Eigenvalues of the Jacobi and SOR iteration matrices and two
worked examples 275
The optimum acceleration parameter for the SOR method. A
necessary theorem 277
Proof of (X+o) l)2 = A.o 2fi2 for block tridiagonal coefficient
matrices 279
Non zero eigenvalues of the Jacobi iteration matrix 280
Theoretical determination of the optimum relaxation parame¬
ter (ob 282
Calculation of a b for a rectangle and other solution domains 285
The Gauss Seidel iteration matrix H(l) 285
Re ordering of equations and unknowns 286
Point iterative methods and re orderings 287
Introduction to 2 cyclic matrices and consistent ordering 288
2 cyclic matrices 289
Ordering vectors for 2 cyclic matrices 290
Consistent ordering of a 2 cyclic matrix 292
The ordering vector for a block tridiagonal matrix 294
An example of a consistently ordered 2 cyclic matrix that is
not block tridiagonal 297
Additional comments on consistent ordering and the SOR
method 297
Consistent orderings associated with the five point approxima¬
tion to Poisson s equation 298
Stone s strongly implicit iterative method 302
A recent direct method 309
Exercises and solutions 311
INDEX 334
|
| any_adam_object | 1 |
| author | Smith, Gordon D. |
| author_GND | (DE-588)1158286333 |
| author_facet | Smith, Gordon D. |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Third edition |
| format | Book |
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| id | DE-604.BV003479063 |
| illustrated | Not Illustrated |
| indexdate | 2024-11-25T17:10:36Z |
| institution | BVB |
| isbn | 0198596413 0198596502 |
| language | English |
| lccn | 85019867 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-002203995 |
| oclc_num | 12612686 |
| open_access_boolean | |
| owner | DE-384 DE-703 DE-739 DE-355 DE-BY-UBR DE-29T DE-91 DE-BY-TUM DE-91G DE-BY-TUM DE-634 DE-83 DE-188 DE-706 DE-521 |
| owner_facet | DE-384 DE-703 DE-739 DE-355 DE-BY-UBR DE-29T DE-91 DE-BY-TUM DE-91G DE-BY-TUM DE-634 DE-83 DE-188 DE-706 DE-521 |
| physical | xi, 337 Seiten Diagramme |
| publishDate | 1985 |
| publishDateSearch | 1985 |
| publishDateSort | 1985 |
| publisher | Clarendon Press |
| record_format | marc |
| series2 | Oxford applied mathematics and computing science series |
| spellingShingle | Smith, Gordon D. Numerical solution of partial differential equations finite difference methods Equations aux dérivées partielles ram Difference equations Numerical solutions Differential equations, Partial Numerical solutions Finite-Differenzen-Methode (DE-588)4194626-1 gnd Differenzenverfahren (DE-588)4134362-1 gnd Numerische Mathematik (DE-588)4042805-9 gnd Differenzengleichung (DE-588)4012264-5 gnd Differentialgleichung (DE-588)4012249-9 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| subject_GND | (DE-588)4194626-1 (DE-588)4134362-1 (DE-588)4042805-9 (DE-588)4012264-5 (DE-588)4012249-9 (DE-588)4128130-5 (DE-588)4044779-0 |
| title | Numerical solution of partial differential equations finite difference methods |
| title_auth | Numerical solution of partial differential equations finite difference methods |
| title_exact_search | Numerical solution of partial differential equations finite difference methods |
| title_full | Numerical solution of partial differential equations finite difference methods G. D. Smith, Brunel University |
| title_fullStr | Numerical solution of partial differential equations finite difference methods G. D. Smith, Brunel University |
| title_full_unstemmed | Numerical solution of partial differential equations finite difference methods G. D. Smith, Brunel University |
| title_short | Numerical solution of partial differential equations |
| title_sort | numerical solution of partial differential equations finite difference methods |
| title_sub | finite difference methods |
| topic | Equations aux dérivées partielles ram Difference equations Numerical solutions Differential equations, Partial Numerical solutions Finite-Differenzen-Methode (DE-588)4194626-1 gnd Differenzenverfahren (DE-588)4134362-1 gnd Numerische Mathematik (DE-588)4042805-9 gnd Differenzengleichung (DE-588)4012264-5 gnd Differentialgleichung (DE-588)4012249-9 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| topic_facet | Equations aux dérivées partielles Difference equations Numerical solutions Differential equations, Partial Numerical solutions Finite-Differenzen-Methode Differenzenverfahren Numerische Mathematik Differenzengleichung Differentialgleichung Numerisches Verfahren Partielle Differentialgleichung |
| url | http://www.loc.gov/catdir/enhancements/fy0603/85019867-d.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002203995&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT smithgordond numericalsolutionofpartialdifferentialequationsfinitedifferencemethods |