Contents Preface Ill About the Authors V 1 Difference methods for the Fisher equation 1 1.1 Introduction 1 1.2 Notation and lemmas 3 1.3 Forward Euler difference scheme 11 1.3.1 Derivation of the difference scheme 11 1.3.2 Solvability and convergence of the difference scheme 12 1.4 Backward Euler difference scheme 14 1.4.1 Derivation of the difference scheme 14 1.4.2 Existence and convergence of the difference solution 16 1.5 Crank-Nicolson difference scheme 18 1.5.1 Derivation of the difference scheme 19 1.5.2 Existence and convergence of the difference solution 20 1.6 Fourth-order compact difference scheme 22 1.6.1 Derivation of the difference scheme 22 1.6.2 Existence and convergence of difference solution 23 1.7 Three-level linearized difference scheme 27 1.7.1 Derivation of the difference scheme 27 1.7.2 Existence and convergence of the difference solution 31 1.8 Numerical experiments 36 1.9 Summary and extension 38 2 Difference methods for the Burgers' equation 41 2.1 Introduction 41 2.2 Two-level nonlinear difference scheme 43 2.2.1 Derivation of the difference scheme 43 2.2.2 Conservation and boundedness of the difference solution 44 2.2.3 Existence and uniqueness of the difference solution 46 2.2.4 Convergence of the difference solution 49 2.3 Three-level linearized difference scheme 54 2.3.1 Derivation of the difference scheme 54 2.3.2 Existence and uniqueness of the difference solution 55 2.3.3 Conservation and boundedness of the difference solution 56 2.3.4 Convergence of the difference solution 57 2.4 Hopf-Cole transformation and fourth-order difference scheme 61 2.4.1 Hopf-Cole transformation 61 2.4.2 Derivation of the difference scheme 63 2.4.3 Existence and uniqueness of the difference solution 65 2.4.4 Convergence of the difference solution 67 2.4.5 Calculation of the solution of the original problem 69 2.5 Fourth-order compact two-level nonlinear difference scheme 70 2.5.1 Derivation of the difference scheme 73 2.5.2 Conservation and boundedness of the difference solution 74 2.5.3 Existence and uniqueness of the difference solution 75 2.5.4 Convergence of the difference solution 79 2.5.5 Stability of the difference solution 84 2.6 Numerical experiments 87 2.7 Summary and extension 88 3 Difference methods for the regularized long-wave equation 91 3.1 Introduction 91 3.2 Two-level nonlinear difference scheme 92 3.2.1 Derivation of the difference scheme 92 3.2.2 Existence of the difference solution 93 3.2.3 Conservation and boundedness of the difference solution 94 3.2.4 Uniqueness of the difference solution 95 3.2.5 Convergence of the difference solution 96 3.3 Three-level linearized difference scheme 98 3.3.1 Derivation of the difference scheme 98 3.3.2 Conservation and boundedness of the difference solution 99 3.3.3 Existence and uniqueness of the difference solution 100 3.3.4 Convergence of the difference solution 100 3.4 Numerical experiments 103 3.5 Summary and extension 104 4 Difference methods for the Korteweg-de Vries equation 106 4.1 Introduction 106 4.2 First-order in space two-level nonlinear difference scheme 107 4.2.1 Derivation of the difference scheme 107 4.2.2 Existence of the difference solution 110 4.2.3 Conservation and boundedness of the difference solution 112 4.2.4 Convergence of the difference solution 113 4.3 First-order in space three-level linearized difference scheme 115 4.3.1 Derivation of the difference scheme 115 4.3.2 Existence and uniqueness of the difference solution 116 4.3.3 Conservation and boundedness of the difference solution 117 4.3.4 Convergence of the difference solution 118 4.4 Second-order in space two-level nonlinear difference scheme 123 4.4.1 Derivation of the difference scheme 123 4.4.2 Existence of the difference solution 125 4.4.3 Conservation and boundedness of the difference solution 127 4.4.4 Convergence and uniqueness of the difference solution 128 4.5 Second-order in space three-level linearized difference scheme 137 4.5.1 Derivation of the difference scheme 137 4.5.2 Conservation and boundedness of the difference solution 139 4.5.3 Existence and uniqueness of the difference solution 141 4.5.4 Convergence of the difference solution 143 4.6 Numerical experiments 146 4.7 Summary and extension 148 5 Difference methods for the Camassa-Holm equation 151 5.1 Introduction 151 5.2 Two-level nonlinear difference scheme 152 5.2.1 Derivation of the difference scheme 152 5.2.2 Conservation of the difference solution 153 5.2.3 Existence and uniqueness of the difference solution 154 5.2.4 Convergence of the difference solution 157 5.3 Three-level linearized difference scheme 159 5.3.1 Derivation of the difference scheme 159 5.3.2 Conservation and boundedness of the difference solution 160 5.3.3 Existence and uniqueness of the difference solution 161 5.3.4 Convergence of the difference solution 162 5.4 Numerical experiments 168 5.5 Summary and extension 172 6 Difference methods for the Schrodinger equation 174 6.1 Introduction