Strong stability preserving Runge-Kutta and multistep time discretizations

Contents......Page 10
Preface......Page 8
1. Overview: The Development of SSP Methods......Page 14
2.2 Motivation......Page 22
2.3 SSP methods as convex combinations of Euler's method: the Shu-Osher formulation......Page 25
2.4.1 A second order method......Page 28
2.4.2 A third order method......Page 29
2.4.3 A fourth order method......Page 33
3. The SSP Coe cient for Runge–Kutta Methods......Page 38
3.1 The modi ed Shu-Osher form......Page 40
3.1.1 Vector notation......Page 42
3.2 Unique representations......Page 43
3.2.1 The Butcher form......Page 44
3.2.2 Reducibility of Runge{Kutta methods......Page 45
3.3 The canonical Shu-Osher form......Page 46
3.3.1 Computing the SSP coefficient......Page 50
3.4 Formulating the optimization problem......Page 52
3.5 Necessity of the SSP time step restriction......Page 53
4. SSP Runge–Kutta Methods for Linear Constant Coefficient Problems......Page 56
4.1 The circle condition......Page 57
4.2 An example: the midpoint method......Page 58
4.3.1 Formulas for the stability function......Page 59
4.3.2 An alternative form......Page 62
4.4 Strong stability preservation for linear systems......Page 63
4.5 Absolute monotonicity......Page 64
4.6 Necessity of the time step condition......Page 66
4.7 An optimization algorithm......Page 67
4.8 Optimal SSP Runge{Kutta methods for linear problems......Page 69
4.9 Linear constant coe cient operators with time dependent forcing terms......Page 71
5.1 Order barriers......Page 76
5.1.1 Stage order......Page 77
5.1.2 Order barrier for explicit Runge–Kutta methods......Page 78
5.1.3 Order barrier for implicit Runge–Kutta methods......Page 79
5.1.4 Order barriers for diagonally implicit and singly implicit methods......Page 80
5.2 Bounds on the SSP coefficient......Page 82
5.2.1 Bounds for explicit Runge–Kutta methods......Page 84
5.2.2 Unconditional strong stability preservation......Page 87
6. Low Storage Optimal Explicit SSP Runge–Kutta Methods......Page 90
6.1 Low-storage Runge–Kutta algorithms......Page 91
6.1.1 Williamson (2N) methods......Page 92
6.1.2 van der Houwen (2R) methods......Page 93
6.1.3 2S and 2S* methods......Page 94
6.2.1 Second order methods......Page 96
6.2.2 Third order methods......Page 98
6.2.3 Fourth order methods......Page 100
6.3 Embedded optimal SSP pairs......Page 101
7. Optimal Implicit SSP Runge–Kutta Methods......Page 104
7.1 Barriers, bounds, and bonuses......Page 106
7.2 Optimal second order and third order methods......Page 108
7.3 Optimal fourth through sixth order methods......Page 109
7.4.1 Fourth order methods......Page 110
7.4.2 Fifth order methods......Page 113
7.4.3 Sixth order methods......Page 116
8. SSP Properties of Linear Multistep Methods......Page 122
8.1.1 Explicit methods......Page 124
8.1.2 Implicit methods......Page 126
8.2.1 Second order methods......Page 127
8.2.2 Third order methods......Page 129
8.3.2 Implicit methods......Page 130
8.4 Starting methods......Page 132
9.1 SSP theory of general linear methods......Page 136
9.2 Two-step Runge{Kutta methods......Page 139
9.2.1 Conditions and barriers for SSP two-step Runge–Kutta methods......Page 141
9.3.1 Formulating the optimization problem......Page 144
9.3.2 Efficient implementation of Type II SSP TSRKs......Page 145
9.3.3 Optimal methods of orders one to four......Page 146
9.3.4 Optimal methods of orders  ve to eight......Page 147
9.4.1 Fifth order SSP TSRK methods......Page 149
9.4.2 Sixth order SSP TSRK methods......Page 150
9.4.3 Seventh order SSP TSRK methods......Page 151
9.4.4 Eighth order SSP TSRK methods......Page 152
10.1 SSP methods with negative  ij 's......Page 154
10.2.1 Second and third order methods......Page 157
10.2.2 Fourth order methods......Page 159
10.2.3 A fifth order method......Page 160
10.3 Optimal explicit multistep schemes with downwinding......Page 161
10.4 Application: Deferred correction methods......Page 165
11.1 TVD schemes......Page 170
11.2 Maximum principle satisfying schemes and positivity preserving schemes......Page 172
11.3 Coercive approximations......Page 177
Bibliography......Page 180
Index......Page 188