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2.2.5
57
2.3
Continuous and Discrete Solutions ..............
57
2.3.1
Diﬀerence and Diﬀerential Equations .........
Symmetry ........................
2.3.2
58
Uniqueness .......................
61
2.3.3
61
2.3.4
A Maximum Principle for the Discrete Problem . . .
2.3.5
Convergence of the Discrete Solutions ........
63
Eigenvalue Problems ......................
2.4
65
65
2.4.1
The Continuous Eigenvalue Problem .........
2.4.2
68
The Discrete Eigenvalue Problem ...........
Exercises
2.5 Contents Positive Deﬁnite Matrices ............... 55
............................
72
2.6 Projects ............................. 82
3 The Heat Equation 87
3.1 A Brief Overview ........................ 88
3.2 Separation of Variables ..................... 90
3.3 The Principle of Superposition ................ 92
3.4 Fourier Coeﬃcients ....................... 95
3.5 Other Boundary Conditions .................. 97
3.6 The Neumann Problem .................... 98
3.6.1 The Eigenvalue Problem ................ 99
3.6.2 Particular Solutions .................. 100
3.6.3 A Formal Solution ................... 101
3.7 Energy Arguments ....................... 102
3.8 Diﬀerentiation of Integrals ................... 106
3.9 Exercises ............................ 108
3.10 Projects ............................. 113
4 Finite Diﬀerence Schemes For The Heat Equation 117
4.1 An Explicit Scheme ...................... 119
4.2 Fourier Analysis of the Numerical Solution ......... 122
4.2.1 Particular Solutions .................. 123
4.2.2 Comparison of the Analytical and Discrete Solution 127
4.2.3 Stability Considerations ................ 129
4.2.4 The Accuracy of the Approximation ......... 130
4.2.5 Summary of the Comparison ............. 131
4.3 Von Neumann’s Stability Analysis .............. 132
4.3.1 Particular Solutions: Continuous and Discrete .... 133
4.3.2 Examples ........................ 134
4.3.3 A Nonlinear Problem ................. 137
4.4 An Implicit Scheme ....................... 140
4.4.1 Stability Analysis .................... 143
4.5 Numerical Stability by Energy Arguments ......... 145
4.6 Exercises ............................ 148

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