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annotate +time/CdiffImplicit.m @ 379:ca73ee0623e5 feature/beams

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Added an implicit central time stepping scheme.
author Jonatan Werpers <jonatan@werpers.com>
date Fri, 09 Dec 2016 16:03:30 +0100
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children 280ae4dbf93b
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379
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1 classdef CdiffImplicit < time.Timestepper
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2 properties
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3 A, B, C, G
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4 AA, BB, CC
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5 k
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6 t
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7 v, v_prev
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8 n
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9
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10 % LU factorization
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11 L,U,p,q
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12 end
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13
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14 methods
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15 % Solves
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16 % A*u_tt + B*u + C*v_t = G(t)
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17 % u(t0) = f1
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18 % u_t(t0) = f2
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19 % starting at time t0 with timestep k
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20 function obj = CdiffImplicit(A, B, C, G, f1, f2, k, t0)
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21 default_arg('A', []);
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22 default_arg('C', []);
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23 default_arg('G', []);
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24 default_arg('f1', 0);
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25 default_arg('f2', 0);
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26 default_arg('t0', 0);
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27
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28 m = size(B,1);
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29
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30 if isempty(A)
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31 A = speye(m);
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32 end
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33
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34 if isempty(C)
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35 C = sparse(m,m);
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36 end
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37
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38 if isempty(G)
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39 G = @(t) sparse(m,m);
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40 end
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41
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42 if isempty(f1)
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43 f1 = sparse(m,m);
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44 end
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45
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46 if isempty(f2)
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47 f2 = sparse(m,m);
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48 end
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49
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50 obj.A = A;
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51 obj.B = B;
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52 obj.C = C;
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53 obj.G = G;
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54
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55 AA = 1/k^2*A + 1/2*B + 1/(2*k)*C;
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56 BB = -2/k^2*A;
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57 CC = 1/k^2*A + 1/2*B - 1/(2*k)*C;
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58 % AA*v_next + BB*v + CC*v_prev == G(t_n)
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59
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60 obj.AA = AA;
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61 obj.BB = BB;
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62 obj.CC = CC;
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63
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64 v_prev = f1;
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65 I = speye(m);
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66 v = (1/k^2*A)\((1/k^2*A - 1/2*B)*f1 + (1/k*I - 1/2*C)*f2 + 1/2*G(0));
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67
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68 if ~issparse(A) || ~issparse(B) || ~issparse(C)
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69 error('LU factorization with full pivoting only works for sparse matrices.')
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70 end
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71
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72 [L,U,p,q] = lu(AA,'vector');
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73
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74 obj.L = L;
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75 obj.U = U;
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76 obj.p = p;
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77 obj.q = q;
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78
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79
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80 obj.k = k;
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81 obj.t = t0+k;
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82 obj.n = 0;
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83 obj.v = v;
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84 obj.v_prev = v_prev;
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85 end
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86
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87 function [v,t] = getV(obj)
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88 v = obj.v;
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89 t = obj.t;
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90 end
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91
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92 function [vt,t] = getVt(obj)
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93 vt = (obj.v-obj.v_prev)/obj.k;
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94 t = obj.t;
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95 end
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96
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97 function obj = step(obj)
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98 b = obj.G(obj.t) - obj.BB*obj.v - obj.CC*obj.v_prev;
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99
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100 % Backslash
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101 obj.v_prev = obj.v;
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102 obj.v = obj.AA\b;
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103
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104 % % LU with column pivot
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105 % y = obj.L\b(obj.p);
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106 % Qx = U\y;
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107 % v = Qx(q);
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108
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109
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110
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111 obj.t = obj.t + obj.k;
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112 obj.n = obj.n + 1;
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113 end
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114 end
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115 end
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116
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117
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118
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119
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120
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121 %%% Derivation
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122 % syms A B C G
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123 % syms n k
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124 % syms f1 f2
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125
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126 % v = symfun(sym('v(n)'),n);
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127
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128
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129 % d = A/k^2 * (v(n+1) - 2*v(n) +v(n-1)) + B/2*(v(n+1)+v(n-1)) + C/(2*k)*(v(n+1) - v(n-1)) == G
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130 % ic1 = v(0) == f1
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131 % ic2 = A/k*(v(1)-f1) + k/2*(B*f1 + C*f2 - G) - f2 == 0
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132
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133 % c = collect(d, [v(n) v(n-1) v(n+1)]) % (-(2*A)/k^2)*v(n) + (B/2 + A/k^2 - C/(2*k))*v(n - 1) + (B/2 + A/k^2 + C/(2*k))*v(n + 1) == G
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134 % syms AA BB CC
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135 % % AA = B/2 + A/k^2 + C/(2*k)
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136 % % BB = -(2*A)/k^2
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137 % % CC = B/2 + A/k^2 - C/(2*k)
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138 % s = subs(c, [B/2 + A/k^2 + C/(2*k), -(2*A)/k^2, B/2 + A/k^2 - C/(2*k)], [AA, BB, CC])
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139
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140
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141 % ic2_a = collect(ic2, [v(1) f1 f2]) % (A/k)*v(1) + ((B*k)/2 - A/k)*f1 + ((C*k)/2 - 1)*f2 - (G*k)/2 == 0
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142