High Order Strong Stability Preserving Time Integrators and Numerical Wave Propagation Methods for Hyperbolic PDEs David I. Ketcheson A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Washington 2009 Program Authorized to Offer Degree: Applied Mathematics University of Washington Graduate School This is to certify that I have examined this copy of a doctoral dissertation by David I. Ketcheson and have found that it is complete and satisfactory in all respects, and that any and all revisions required by the final examining committee have been made. Chair of the Supervisory Committee: Randall J. LeVeque Reading Committee: Randall J. LeVeque Bernard Deconinck Kenneth Bube Date: In presenting this dissertation in partial fulfillment of the requirements for the doctoral degree at the University of Washington, I agree that the Library shall make its copies freely available for inspection. I further agree that extensive copying of this dissertation is allowable only for scholarly purposes, consistent with “fair use” as prescribed in the U.S. Copyright Law. Requests for copying or reproduction of this dissertation may be referred to Proquest Information and Learning, 300 North Zeeb Road, Ann Arbor, MI 48106-1346, 1-800-521-0600, to whom the author has granted “the right to reproduce and sell (a) copies of the manuscript in microform and/or (b) printed copies of the manuscript made from microform.” Signature Date .. University of Washington Abstract High Order Strong Stability Preserving Time Integrators and Numerical Wave Propagation Methods for Hyperbolic PDEs David I. Ketcheson Chair of the Supervisory Committee: Professor Randall J. LeVeque Applied Mathematics Hyperbolic PDEs describe the great variety of physical phenomena governed by wave behavior. Numerical methods are necessary to provide approximate solutions to real wave propagation problems. High order accurate methods are often essential to achieve accurate solutions on computationally feasible grids. However, maintaining numerical stability and satisfying physical constraints becomes increasingly difficult as higher order methods are employed. In this thesis, high order numerical tools for hyperbolic PDEs are developed in a method of lines framework. Optimal high order accurate strong sta- bility preserving (SSP) time integrators, for both linear and nonlinear systems, are devel- oped. Improved SSP methods are found as a result of rewriting the relevant optimization problems in a form more amenable to efficient solution. A new, very general class of low-storage Runge-Kutta methods is proposed (based on the form of some optimal SSP methods) and shown to include methods with properties that cannot be achieved by existing classes of low-storage methods. A high order accurate semi-discrete wave propa- gation method is developed in one and two dimensions using wave propagation Riemann solvers and high order weighted essentially non-oscillatory (WENO) reconstruction. The space and time discretizations are combined to achieve a high order accurate method for general hyperbolic PDEs. This method is applied to model solitary waves in a nonlinear, non-dispersive periodic heterogeneous medium. TABLE OF CONTENTS Page List of Figures . vi List of Tables . viii Chapter 1: Introduction . .1 1.1 Motivation . .1 1.1.1 High Order Numerical Methods for Hyperbolic Conservation Laws1 1.1.2 Strong Stability Preserving Time Integration . .2 1.1.3 Flux Differencing and Wave Propagation . .3 1.1.4 Waves in Heterogeneous Media . .4 1.2 Outline . .4 Chapter 2: Numerical Methods for the Initial Value Problem . .6 2.1 The Method of Lines . .6 2.2 Linear Multistep Methods . .7 2.3 Runge-Kutta Methods . .8 2.4 General Linear Methods . .9 2.5 Additive Methods . 13 Chapter 3: Strong Stability Preserving Methods and Absolute Monotonicity . 15 3.1 Strong Stability Preservation . 17 i 3.1.1 TVD Time Integration . 17 3.1.2 The Shu-Osher Form . 19 3.1.3 Strong Stability Properties . 20 3.2 Absolute Monotonicity and Strong Stability Preservation for Linear IVPs . 23 3.3 Absolute Monotonicity of Runge-Kutta Methods . 26 3.3.1 The SSP Coefficient . 32 3.4 Unconditional Strong Stability Preservation . 33 3.5 Negative Coefficients and Downwinding . 33 3.6 Optimal SSP Methods . 35 3.6.1 Efficiency . 35 3.6.2 The Relation Between R and ...................... 36 C Chapter 4: Optimal Threshold Factors for Linear Initial Value Problems . 38 4.1 Threshold Factors for General Linear Methods . 39 4.2 ’Tall Tree’ Order Conditions for Explicit General Linear Methods . 40 4.3 An Upper Bound . 40 4.4 Solution Algorithm . 42 4.5 Optimal Threshold Factors for Explicit Methods . 43 4.5.1 One-step methods . 43 4.5.2 One-stage multistep methods . 43 4.5.3 Multistage multistep methods . 45 4.6 Threshold Factors for Methods with Downwinding . 48 4.6.1 Threshold Factors for Downwinded Explicit GLMs . 48 4.6.2 Optimal One-step Methods with Downwinding . 50 Chapter 5: Optimal SSP Linear Multistep Methods . 53 5.1 General Solution Algorithm . 54 5.2 Bounds on the SSP Coefficient . 54 5.2.1 Explicit Methods . 54 5.2.2 Implicit Methods . 55 5.3 Optimal Methods without Downwinding . 55 5.4 Optimal Methods with Downwinding . 56 Chapter 6: SSP Runge-Kutta Methods . 63 6.1 Bounds on the SSP Coefficient for Runge-Kutta Methods . 64 6.2 Formulation of the Optimization Problem . 67 ii 6.3 Implicit Runge-Kutta Methods . 69 6.3.1 Optimal Methods . 70 6.3.2 Numerical Experiments . 76 6.4 Explicit Runge-Kutta Methods . 82 6.4.1 Memory Considerations . 83 6.4.2 Optimal Methods . 84 6.4.3 Absolute Stability Regions . 91 6.4.4 Internal Stability . 91 6.4.5 Truncation Error Analysis . 94 6.4.6 Embedding optimal SSP methods . 95 6.4.7 Numerical Experiments . 96 6.5 Summary and Conjectures . 100 Chapter 7: Low-Storage Runge-Kutta Methods . 103 7.1 Introduction . 103 7.2 Two-register Methods . 105 7.2.1 Williamson (2N) methods . 106 7.2.2 van der Houwen (2R) methods . 107 7.3 Low-Storage Methods Have Sparse Shu-Osher Forms . 108 7.3.1 2N Methods . 108 7.3.2 2R Methods . 109 7.4 2S Methods . 109 7.4.1 2S* Methods . 110 7.4.2 2S Embedded Pairs . 111 7.4.3 3S* Methods . 112 7.5 Feasibility of Low-storage Assumptions . 114 7.5.1 2N Methods . 114 7.5.2 2R Methods . 114 7.5.3 2S Methods . 115 7.6 Improved low-storage methods . 115 7.7 Conclusions . 117 Chapter 8: Numerical Wave Propagation . 118 8.1 Linear Hyperbolic Systems . 118 8.2 The Semi-discrete Wave-Propagation form of Godunov’s Method . 119 8.3 Extension to Higher Order . 122 iii 8.4 Variable Coefficient Linear Systems . 123 8.5 Nonlinear Systems . 124 8.6 High Order Non-oscillatory Reconstruction of Scalar Functions . 125 8.6.1 Linear (Non-limited) Reconstruction . 125 8.6.2 TVD Reconstruction . 126 8.6.3 Weighted Essentially Non-Oscillatory Reconstruction . 127 8.7 Reconstruction of Vector-valued Functions . 128 8.7.1 Reconstruction of Eigencomponent Coefficients . 128 8.7.2 Characteristic-wise Reconstruction . 129 8.7.3 Wave-slope Reconstruction . 130 8.8 Extension to Two Dimensions . 130 Chapter 9: Numerical Tests . 132 9.1 Methods . 132 9.2 Acoustics . 133 9.2.1 Single Material Interface . 134 9.2.2 Several Interfaces . 138 9.2.3 A Sonic Crystal . 138 9.3 Fluid Dynamics . 142 Chapter 10: Stegotons . 144 10.1 Previous Work . ..