xiv
Contents
2.3.2 Single-Stage Two-Level Schemes
50
2.3.3 Multistage Methods . . . . . . .
53
2.3.4 Three-Level Schemes
56
2.3.5 Controlling the Leapfrog Computational Mode
60
2.3.6 Higher-Order Schemes . . . . . .
65
2.4 Space-Differencing . . . . .. . . . . . .
72
2.4.1 Differential-Difference Equations
and Wave Dispersion
73
2.4.2 Dissipation, Dispersion, and the Modified Equation
80
2.4.3 Artificial Dissipation . . . . . . .
82
2.4.4 Compact Differencing . . . . . .
86
2. 2.5.1 5.1 The The D Dis isc cr rete-D ete-Disp isper ers sion ion Relat Relatio ion n
91 91
2. 2.5.2 5.2 The The Modifi Modified ed E Eq qu ua at ti io on n. . . . . . . . . . . .
94 94
2. 2.5.3 5.3 The The Lax-W Lax-Wen endroff droff Method Method . . . . . .
95 95
2.6 Summary Discussion of Elementary Methods
99
Problems . . . . . . . . . . . . . . . . . . . . . .
10I
3 Beyend the ODe-Way Wave Equation
107
3.1 Systems of Equations ..
107
3.1.1 Stability .. . . . . . . .
108
3.1.2 Staggered Meshes . . . .
113
3.2.2
3. 3.3.1 3.1
3. 3.3.2 3.2
3. 3.3.3 3.3
3. 3.4.1 4.1
3. 3.4.2 4.2
3. 3.4.3 4.3
Systems of Equations in Several Dimensions.
Spli Split t Expli Explicit cit Sch Schem emes es . . . .
S Sp plit lit Imp Implicit licit S Sch che emes mes . . . .
Sta Stability bility o of f S Split plit Sch Schemes emes
Pure Pure Diffus Diffusio ion n . . . . . . . . . . . .
Adve Advecti ctio on n and and Diff Diffu usio sion n . .
Adve Advecti ctio on n with with S Sources ources an and d Sin Sinks ks
126
13 130 0
13 132 2
13 134 4
13 136 6
13 138 8
14 144 4
3.5 Linear Equations with Variable Coefficients
3.5.1 Aliasing Error . . . . . .
3.5.2 Conservation and Stability
147
148
154
3. 3.6.1 6.1 Bur Burg ger ers's s's E Eq qu ua ation tion . . . . . . . .
15 159 9
3. 3.6.2 6.2 T Th he e Barotrop Barotropic ic V Vo or rt ti ic ci it ty y Equa Equat ti ion on
16 163 3
Problems . . . . . . . . . . . . . . . . . . . .
167