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First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
author Martin Almquist <malmquist@stanford.edu>
date Sun, 11 Apr 2021 21:10:23 +0200
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children 56439c1a49b4
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3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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1 classdef Laplace1dVariable < scheme.Scheme
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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2 properties
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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3 grid
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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4 order % Order accuracy for the approximation
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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5
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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6 D % scheme operator
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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7 H % Discrete norm
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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8
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9 % Variable coefficients a(b u_x)_x
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10 a
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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11 b
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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12
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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13 Hi
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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14 e_l
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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15 e_r
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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16 d_l
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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17 d_r
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18 gamm
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19 end
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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20
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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21 methods
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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22 function obj = Laplace1dVariable(g, order, a, b)
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23 default_arg('a', @(x) 0*x + 1);
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24 default_arg('b', @(x) 0*x + 1);
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25
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26 assertType(g, 'grid.Cartesian');
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27
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28 ops = sbp.D2Variable(g.size(), g.lim{1}, order);
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29
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30 obj.H = sparse(ops.H);
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31 obj.Hi = sparse(ops.HI);
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32 obj.e_l = sparse(ops.e_l);
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33 obj.e_r = sparse(ops.e_r);
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34 obj.d_l = -sparse(ops.d1_l);
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35 obj.d_r = sparse(ops.d1_r);
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36
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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37
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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38 obj.grid = g;
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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39 obj.order = order;
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40
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41 a = grid.evalOn(g, a);
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42 b = grid.evalOn(g, b);
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43
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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44 A = spdiag(a);
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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45 B = spdiag(b);
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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46
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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47 obj.a = A;
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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48 obj.b = B;
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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49
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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50 obj.D = A*ops.D2(b);
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51
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52 obj.gamm = g.h*ops.borrowing.M.d1;
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53 end
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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54
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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55
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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56 % Closure functions return the opertors applied to the own doamin to close the boundary
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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57 % Penalty functions return the opertors to force the solution. In the case of an interface it returns the operator applied to the other doamin.
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58 % boundary is a string specifying the boundary e.g. 'l','r' or 'e','w','n','s'.
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59 % type is a string specifying the type of boundary condition if there are several.
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60 % data is a function returning the data that should be applied at the boundary.
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61 % neighbour_scheme is an instance of Scheme that should be interfaced to.
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62 % neighbour_boundary is a string specifying which boundary to interface to.
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63 function [closure, penalty] = boundary_condition(obj,boundary,type,data)
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64 default_arg('type','neumann');
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65 default_arg('data',0);
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66
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67 e = obj.getBoundaryOperator('e', boundary);
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68 d = obj.getBoundaryOperator('d', boundary);
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69 s = obj.getBoundarySign(boundary);
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70
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71 b = e'*obj.b*e;
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72
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73 switch type
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74 % Dirichlet boundary condition
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75 case {'D','d','dirichlet'}
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76 tuning = 1.1;
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77 tau1 = -tuning/obj.gamm;
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78 tau2 = 1;
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79
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80 tau = tau1*e + tau2*d;
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81
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82 closure = obj.a*obj.Hi*tau*b*e';
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83 penalty = obj.a*obj.Hi*tau*b;
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84
3f4e011193f0 First implementation of Laplace1dVariable. Passes symmetry and definiteness tests.
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85 % Neumann boundary condition
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86 case {'N','n','neumann'}
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87 tau = -e;
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88
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89 closure = obj.a*obj.Hi*tau*b*d';
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90 penalty = -obj.a*obj.Hi*tau*b;
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91
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92 % Unknown, boundary condition
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93 otherwise
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94 error('No such boundary condition: type = %s',type);
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95 end
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96 end
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97
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98 function [closure, penalty] = interface(obj, boundary, neighbour_scheme, neighbour_boundary, type)
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99 % u denotes the solution in the own domain
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100 % v denotes the solution in the neighbour domain
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101 e_u = obj.getBoundaryOperator('e', boundary);
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102 d_u = obj.getBoundaryOperator('d', boundary);
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103 s_u = obj.getBoundarySign(boundary);
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104
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105 e_v = neighbour_scheme.getBoundaryOperator('e', neighbour_boundary);
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106 d_v = neighbour_scheme.getBoundaryOperator('d', neighbour_boundary);
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107 s_v = neighbour_scheme.getBoundarySign(neighbour_boundary);
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108
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109 b_u = e_u'*obj.b*e_u;
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110 b_v = e_v'*neighbour_scheme.b*e_v;
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111
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112 gamm_u = obj.gamm;
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113 gamm_v = neighbour_scheme.gamm;
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114
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115 tuning = 1.1;
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116
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117 closure = zeros(size(obj.D));
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118 penalty = zeros(length(obj.D), length(b_v));
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119
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120 % Continuity of bu_x
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121 closure = closure - 1/2*e_u*b_u*d_u';
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122 penalty = penalty - 1/2*e_u*b_v*d_v';
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123
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124 % Continuity of u (symmetrizing term)
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125 closure = closure + 1/2*d_u*b_u*e_u';
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126 penalty = penalty - 1/2*d_u*b_u*e_v';
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127
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128 % Continuity of u (symmetric term)
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129 tau = 1/4*(b_u/gamm_u + b_v/gamm_v)*tuning;
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130 closure = closure - e_u*tau*e_u';
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131 penalty = penalty + e_u*tau*e_v';
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132
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133 % Multiply by Hi and a.
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134 closure = obj.Hi*obj.a*closure;
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135 penalty = obj.Hi*obj.a*penalty;
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136
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137 end
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138
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139 % Returns the boundary operator op for the boundary specified by the string boundary.
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140 % op -- string
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141 % boundary -- string
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142 function o = getBoundaryOperator(obj, op, boundary)
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143 assertIsMember(op, {'e', 'd'})
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144 assertIsMember(boundary, {'l', 'r'})
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145
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146 o = obj.([op, '_', boundary]);
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147 end
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148
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149 % Returns square boundary quadrature matrix, of dimension
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150 % corresponding to the number of boundary points
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151 %
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152 % boundary -- string
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153 % Note: for 1d diffOps, the boundary quadrature is the scalar 1.
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154 function H_b = getBoundaryQuadrature(obj, boundary)
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155 assertIsMember(boundary, {'l', 'r'})
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156
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157 H_b = 1;
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158 end
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159
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160 % Returns the boundary sign. The right boundary is considered the positive boundary
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161 % boundary -- string
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162 function s = getBoundarySign(obj, boundary)
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163 assertIsMember(boundary, {'l', 'r'})
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164
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165 switch boundary
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166 case {'r'}
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167 s = 1;
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168 case {'l'}
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169 s = -1;
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170 end
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171 end
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172
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173 function N = size(obj)
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174 N = obj.grid.size();
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175 end
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176
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177 end
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178 end