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7.21 Diagram showing the position of the isotherm with temperature u = 0.5 at time t = 0.3461 using a time-step of 0.001 in example 7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 7.22 Diagram showing the position of the freezing front with temperature u = 1 at time t = 0.3461 using a time-step of 0.001 in example 7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 141 8.1 Diagram showing Flynn’s Taxonomy where PU represents a processing unit . . . . . . . . . . . . . . . . . . . . . . . . . . 148 8.2 Diagram showing a shared memory system . . . . . . . . . . 150 8.3 Diagram showing a distributed memory system . . . . . . . . 151 8.4 Speed-up for the method of fundamental solutions. . . . . . . 157 8.5 Speed-up for the Laplace transform isotherm migration method in one dimension allocating Stehfest values to varying num- bers of processors using 21 isotherms in example 8.1 . . . . . 160 8.6 Speed-up for the Laplace transform isotherm migration method in one dimension allocating Stehfest values to varying num- bers of processors and 41 isotherms in example 8.2 . . . . . . 161 8.7 Speed-up for the Laplace transform isotherm migration method in one dimension with no phase change with 21 isotherms in example 8.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 8.8 Speed-up for the Laplace transform isotherm migration method in one dimension for a Stefan problem in example 8.4 . . . . 164 8.9 Speed-up for the Laplace transform isotherm migration method for a two-dimensional Stefan problem in example 8.5 . . . . . 166 xiv
List of Tables 3.1 The position of isotherm 2 with increasing time for example 3.1 42 3.2 The position of isotherm 8 with increasing time for example 3.1 42 3.3 Error in value of ũ at ˜t = 0.1 for example 3.2 . . . . . . . . . 46 3.4 Error in value of ũ at ˜t = 0.5 for example 3.2 . . . . . . . . . 50 3.5 Effect of errors in starting values for isotherm 2 in example 3.3 51 3.6 Percentage errors in solution for isotherm 2 in example 3.3 . . 52 3.7 Effect of errors in starting values for isotherm 8 in example 3.3 53 3.8 Percentage errors in solution for isotherm 8 in example 3.3 . . 54 3.9 The positions of isotherm 2 for linear variation in α in example 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.10 The positions of isotherm 8 for linear variation in α in example 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.11 The positions of isotherms 2 and 8 when an average value of β is taken to find starting values for β = 0.1 in example 3.5 . 57 3.12 The positions of isotherms 2 and 8 when an average value of β is taken to find starting values for β = 5 in example 3.5 . . 57 4.1 Stehfest weights for M = 4,6,8,10,12,14 and 16 . . . . . . . 64 4.2 Comparison of the results using the isotherm migration method alone and the Laplace transform with the isotherm migration method for isotherm 2 in example 4.1 . . . . . . . . . . . . . 70 xv
Page 1 and 2: Aspects of the Laplace transform is
Page 3 and 4: Acknowledgements It is my greatest
Page 5 and 6: Abstract There are many different m
Page 7 and 8: Contents 1 Introduction 1 2 The con
Page 9 and 10: 4.7.1 Contribution . . . . . . . .
Page 11 and 12: 9.2.3 To extend the use of Laplace
Page 13 and 14: 3.11 The positions of isotherm 8 fo
Page 15: 7.10 Diagram showing the position o
Page 19 and 20: 6.1 The positions of the freezing f
Page 21 and 22: u temperature ũ dimensionless u co
Page 23 and 24: Chapter 1 Introduction We begin our
Page 25 and 26: system. In 1998 the Department was
Page 27 and 28: at a point per unit area. If we hav
Page 29 and 30: where G(x,y) is some function indep
Page 31 and 32: condition may vary from one point o
Page 33 and 34: We can find the solution in differe
Page 35 and 36: this is the initial condition. The
Page 37 and 38: Figure 2.2: Typical grid for the fi
Page 39 and 40: fundamental solution which satisifi
Page 41 and 42: tending the technique to non-homoge
Page 43 and 44: 2.2.7 The method of separation of v
Page 45 and 46: to a set of linear equations of the
Page 47 and 48: labelled Qk, where k = 1,2,...,n
Page 49 and 50: in the case of melting ice (Crank a
Page 51 and 52: and the moving boundary differs. In
Page 53 and 54: 2.3.5 Fixed-domain methods In certa
Page 55 and 56: 2.4 Summary of Chapter 2 In this ch
Page 57 and 58: e easily evaluated, topological cha
Page 59 and 60: We wish to write the heat flow equa
Page 61 and 62: Equations (3.8), (3.9), (3.10) and
Page 63 and 64: In this example we consider the cas
Page 65 and 66: and ˜x 0.8 0.7 0.6 0.5 0.4 0.3 0.2
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˜x 0.5 0.4 0.3 0.2 0.1 Isotherm 0
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w 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0
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ũ 10 9 8 7 6 5 0 0.2 0.4 0.6 0.8 1
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Table 3.5: Effect of errors in star
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Table 3.7: Effect of errors in star
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˜x 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
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it is much more straightforward to
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Chapter 4 The Laplace transform iso
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available in tables. We do have an
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N� A = − 3. Gauss quadrature, w
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The value of M is chosen by the use
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are found where wj are the Stehfest
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where h is the difference in the va
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Table 4.3: Comparison of the result
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α = 1 + u. For the steady-state so
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Table 4.5: The positions of the iso
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To start this problem, we use an ac
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4.7 Summary of Chapter 4 In this ch
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differences, and we shall show how
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˜x = 0, ũ = ũL, ˜t � 0 As in
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However, we find that doing this ca
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front and isotherm 8 respectively u
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straightforward and might require o
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freezing front is given by x0 = 2φ
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x 1 0.8 0.6 0.4 0.2 Freezing front
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ũ(˜x, ˜t) = 0, ˜x = ˜x0, ˜t
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so that when X = 0 Eliminating ũw
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˜x 1 0.8 0.6 0.4 0.2 Steady Convec
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˜x 1 0.8 0.6 0.4 0.2 Steady Convec
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Page 127 and 128:
∂u ∂y = 0 y = 0 0 � x � 1 t
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6.3 The finite difference form We e
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Results We see from table 6.1 that
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where t0 is the initial time at whi
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x 1 0.8 0.6 0.4 0.2 Freezing front
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Page 139 and 140:
grid line x = xM = 1 for all t > 0.
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7.2 Solidification of a square pris
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which is our previous equation (6.8
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Figure 7.2: Diagram showing possibl
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Poots (1962) defines a function �
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y 1 0.8 0.6 0.4 0.2 t=0.05 t=0.1 t=
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so that we have ¯y (n) = y (t0) λ
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y 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0
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y 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0
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Table 7.3: Values of the y co-ordin
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y 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6
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y 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6
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y 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.
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espaced using a given algorithm. We
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Chapter 8 The use of multiple proce
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In considering parallel computing m
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The issue with shared memory system
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Figure 8.3: Diagram showing a distr
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are several MPI implementations ava
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communication in the program adds e
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Sp 4 3.5 3 2.5 2 1.5 1 1 1.5 2 2.5
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work. Because the calculation invol
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Table 8.3: Speed-up when allocating
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not occur when only one process is
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Therefore, as in examples 8.1 and 8
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upon which we focus. We mention the
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heating of a rod at zero degrees, n
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described by an equation involving
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grid points. An example, described
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simple. 9.2 Research objectives Our
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Laplace transform, new complexities
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orthogonal flow lines. To implement
Page 203 and 204:
Crank J (1957) Two methods for the
Page 205 and 206:
Davies A J and Mushtaq J (1998) Par
Page 207 and 208:
Kreider D L, Kuller R G, Ostberg D
Page 209 and 210:
Price R H and Slack M R (1954) The
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