Table of Contents
Preface vii
1 Overview: The Development of SSP Methods 1
2 Strong Stability Preserving Explicit Runge-Kutta Methods 9
2.1 Overview 9
2.2 Motivation 9
2.3 SSP methods as convex combinations of Euler's method: the Shu-Osher formulation 12
2.4 Some optimal SSP Runge-Kutta methods 15
2.4.1 A second order method 15
2.4.2 A third order method 16
2.4.3 A fourth order method 20
3 The SSP Coefficient for Runge-Kutta Methods 25
3.1 The modified Shu-Osher form 27
3.1.1 Vector notation 29
3.2 Unique representations 30
3.2.1 The Butcher form 31
3.2.2 Reducibility of Runge-Kutta methods 32
3.3 The canonical Shu-Osher form 33
3.3.1 Computing the SSP coefficient 37
3.4 Formulating the optimization problem 39
3.5 Necessity of the SSP time step restriction 40
4 SSP Runge-Kutta Methods for Linear Constant Coefficient Problems 43
4.1 The circle condition 44
4.2 An example: the midpoint method 45
4.3 The stability function 46
4.3.1 Formulas for the stability function 46
4.3.2 An alternative form 49
4.3.3 Order conditions on the stability function 50
4.4 Strong stability preservation for linear systems 50
4.5 Absolute monotonicity 51
4.6 Necessity of the time step condition 53
4.7 An optimization algorithm 54
4.8 Optimal SSP Runge-Kutta methods for linear problems 56
4.9 Linear constant coefficient operators with time dependent forcing terms 58
5 Bounds and Barriers for SSP Runge-Kutta Methods 63
5.1 Order barriers 63
5.1.1 Stage order 64
5.1.2 Order barrier for explicit Runge-Kutta methods 65
5.1.3 Order barrier for implicit Runge-Kutta methods 66
5.1.4 Order barriers for diagonally implicit and singly implicit methods 67
5.2 Bounds on the SSP coefficient 69
5.2.1 Bounds for explicit Runge-Kutta methods 71
5.2.2 Unconditional strong stability preservation 74
6 Low Storage Optimal Explicit SSP Runge-Kutta Methods 77
6.1 Low-storage Runge-Kutta algorithms 78
6.1.1 Williamson (2N) methods 79
6.1.2 van der Houwen (2R) methods 80
6.1.3 2S and 2S* methods 81
6.2 Optimal SSP low-storage explicit Runge-Kutta methods 83
6.2.1 Second order methods 83
6.2.2 Third order methods 85
6.2.3 Fourth order methods 87
6.3 Embedded optimal SSP pairs 88
7 Optimal Implicit SSP Runge-Kutta Methods 91
7.1 Barriers, bounds, and bonuses 93
7.2 Optimal second order and third order methods 95
7.3 Optimal fourth through sixth order methods 96
7.4 Coefficients of optimal implicit SSP Runge-Kutta methods 97
7.4.1 Fourth order methods 97
7.4.2 Fifth order methods 100
7.4.3 Sixth order methods 103
8 SSP Properties of Linear Multistep Methods 109
8.1 Bounds and barriers 111
8.1.1 Explicit methods 111
8.1.2 Implicit methods 113
8.2 Explicit SSP multistep methods using few stages 114
8.2.1 Second order methods 114
8.2.2 Third order methods 116
8.2.3 Fourth order methods 117
8.3 Optimal methods of higher order and more steps 117
8.3.1 Explicit methods 117
8.3.2 Implicit methods 117
8.4 Starting methods 119
9 SSP Properties of Multistep Multi-Stage Methods 123
9.1 SSP theory of general linear methods 123
9.2 Two-step Runge-Kutta methods 126
9.2.1 Conditions and barriers for SSP two-step Runge-Kutta methods 128
9.3 Optimal two-step Runge-Kutta methods 131
9.3.1 Formulating the optimization problem 131
9.3.2 Efficient implementation of Type II SSP TSRKs 132
9.3.3 Optimal methods of orders one to four 133
9.3.4 Optimal methods of orders five to eight 134
9.4 Coefficients of optimal methods 136
9.4.1 Fifth order SSP TSRK methods 136
9.4.2 Sixth order SSP TSRK methods 137
9.4.3 Seventh order SSP TSRK methods 138
9.4.4 Eighth order SSP TSRK methods 139
10 Downwinding 141
10.1 SSP methods with negative βij's 141
10.2 Explicit SSP Runge-Kutta methods with downwinding 144
10.2.1 Second and third order methods 144
10.2.2 Fourth order methods 146
10.2.3 A fifth order method 147
10.3 Optimal explicit multistep schemes with downwinding 148
10.4 Application: Deferred correction methods 152
11 Applications 157
11.1 TVD schemes 157
11.2 Maximum principle satisfying schemes and positivity preserving schemes 159
11.3 Coercive approximations 164
Bibliography 167
Index 175
