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8 The use of multiple processors to solve diffusion problems using the Laplace transform isotherm migration method 145 8.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 8.1.1 Parallel Programming Models . . . . . . . . . . . . . . 151 8.1.2 MPI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 8.1.3 Designing Parallel Programs . . . . . . . . . . . . . . 153 8.1.4 Communication . . . . . . . . . . . . . . . . . . . . . . 154 8.1.5 Measuring performance . . . . . . . . . . . . . . . . . 154 8.2 Application to the Laplace transform isotherm migration method155 8.2.1 An earlier problem . . . . . . . . . . . . . . . . . . . . 155 8.2.2 The options for parallel implementation of the Laplace transform isotherm migration method . . . . . . . . . 158 8.3 A Stefan problem in one dimension. . . . . . . . . . . . . . . 163 8.4 A Stefan problem in two dimensions . . . . . . . . . . . . . . 164 8.5 Conclusions regarding the Laplace transform isotherm migration method solution in a parallel environment . . . . . . . . 165 8.6 Summary of Chapter 8 . . . . . . . . . . . . . . . . . . . . . . 166 8.6.1 Contribution . . . . . . . . . . . . . . . . . . . . . . . 167 9 Conclusions and future work 168 9.1 Summary of thesis . . . . . . . . . . . . . . . . . . . . . . . . 168 9.2 Research objectives . . . . . . . . . . . . . . . . . . . . . . . . 175 9.2.1 To look at the isotherm migration method to establish its advantages and disadvantages . . . . . . . . . . . . 175 9.2.2 To develop the method further by the use of the Laplace transform method and to test this method by solving problems in one dimension which did not involve phase change . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 viii
9.2.3 To extend the use of Laplace transform isotherm migration method by solving phase change problems both in one and two dimensions . . . . . . . . . . . . . . . . 176 9.2.4 To examine the performance of the Laplace transform isotherm migration method in a parallel environment to find out whether there was a suitable and sensible way of division of work and to establish whether there was any benefit to be gained from the use of a parallel environment . . . . . . . . . . . . . . . . . . . . . . . . 177 9.3 Published work . . . . . . . . . . . . . . . . . . . . . . . . . . 177 9.4 Future research work . . . . . . . . . . . . . . . . . . . . . . . 178 10 References 180 ix
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: 4.7.1 Contribution . . . . . . . .
Page 13 and 14: 3.11 The positions of isotherm 8 fo
Page 15 and 16: 7.10 Diagram showing the position o
Page 17 and 18: List of Tables 3.1 The position of
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
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In this example we consider the cas
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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
Page 85 and 86:
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
Page 125 and 126:
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
Page 137 and 138:
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
Page 157 and 158:
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.
Page 165 and 166:
espaced using a given algorithm. We
Page 167 and 168:
Chapter 8 The use of multiple proce
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In considering parallel computing m
Page 171 and 172:
The issue with shared memory system
Page 173 and 174:
Figure 8.3: Diagram showing a distr
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are several MPI implementations ava
Page 177 and 178:
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
Page 185 and 186:
not occur when only one process is
Page 187 and 188:
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
Page 197 and 198:
simple. 9.2 Research objectives Our
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Laplace transform, new complexities
Page 201 and 202:
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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