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comparison +scheme/Laplace1dVariable.m @ 1323:3f4e011193f0 feature/laplace1d_variable

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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
parents
children 56439c1a49b4
comparison
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1301:8978521b0f06 1323:3f4e011193f0
1 classdef Laplace1dVariable < scheme.Scheme
2 properties
3 grid
4 order % Order accuracy for the approximation
5
6 D % scheme operator
7 H % Discrete norm
8
9 % Variable coefficients a(b u_x)_x
10 a
11 b
12
13 Hi
14 e_l
15 e_r
16 d_l
17 d_r
18 gamm
19 end
20
21 methods
22 function obj = Laplace1dVariable(g, order, a, b)
23 default_arg('a', @(x) 0*x + 1);
24 default_arg('b', @(x) 0*x + 1);
25
26 assertType(g, 'grid.Cartesian');
27
28 ops = sbp.D2Variable(g.size(), g.lim{1}, order);
29
30 obj.H = sparse(ops.H);
31 obj.Hi = sparse(ops.HI);
32 obj.e_l = sparse(ops.e_l);
33 obj.e_r = sparse(ops.e_r);
34 obj.d_l = -sparse(ops.d1_l);
35 obj.d_r = sparse(ops.d1_r);
36
37
38 obj.grid = g;
39 obj.order = order;
40
41 a = grid.evalOn(g, a);
42 b = grid.evalOn(g, b);
43
44 A = spdiag(a);
45 B = spdiag(b);
46
47 obj.a = A;
48 obj.b = B;
49
50 obj.D = A*ops.D2(b);
51
52 obj.gamm = g.h*ops.borrowing.M.d1;
53 end
54
55
56 % Closure functions return the opertors applied to the own doamin to close the boundary
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.
58 % boundary is a string specifying the boundary e.g. 'l','r' or 'e','w','n','s'.
59 % type is a string specifying the type of boundary condition if there are several.
60 % data is a function returning the data that should be applied at the boundary.
61 % neighbour_scheme is an instance of Scheme that should be interfaced to.
62 % neighbour_boundary is a string specifying which boundary to interface to.
63 function [closure, penalty] = boundary_condition(obj,boundary,type,data)
64 default_arg('type','neumann');
65 default_arg('data',0);
66
67 e = obj.getBoundaryOperator('e', boundary);
68 d = obj.getBoundaryOperator('d', boundary);
69 s = obj.getBoundarySign(boundary);
70
71 b = e'*obj.b*e;
72
73 switch type
74 % Dirichlet boundary condition
75 case {'D','d','dirichlet'}
76 tuning = 1.1;
77 tau1 = -tuning/obj.gamm;
78 tau2 = 1;
79
80 tau = tau1*e + tau2*d;
81
82 closure = obj.a*obj.Hi*tau*b*e';
83 penalty = obj.a*obj.Hi*tau*b;
84
85 % Neumann boundary condition
86 case {'N','n','neumann'}
87 tau = -e;
88
89 closure = obj.a*obj.Hi*tau*b*d';
90 penalty = -obj.a*obj.Hi*tau*b;
91
92 % Unknown, boundary condition
93 otherwise
94 error('No such boundary condition: type = %s',type);
95 end
96 end
97
98 function [closure, penalty] = interface(obj, boundary, neighbour_scheme, neighbour_boundary, type)
99 % u denotes the solution in the own domain
100 % v denotes the solution in the neighbour domain
101 e_u = obj.getBoundaryOperator('e', boundary);
102 d_u = obj.getBoundaryOperator('d', boundary);
103 s_u = obj.getBoundarySign(boundary);
104
105 e_v = neighbour_scheme.getBoundaryOperator('e', neighbour_boundary);
106 d_v = neighbour_scheme.getBoundaryOperator('d', neighbour_boundary);
107 s_v = neighbour_scheme.getBoundarySign(neighbour_boundary);
108
109 b_u = e_u'*obj.b*e_u;
110 b_v = e_v'*neighbour_scheme.b*e_v;
111
112 gamm_u = obj.gamm;
113 gamm_v = neighbour_scheme.gamm;
114
115 tuning = 1.1;
116
117 closure = zeros(size(obj.D));
118 penalty = zeros(length(obj.D), length(b_v));
119
120 % Continuity of bu_x
121 closure = closure - 1/2*e_u*b_u*d_u';
122 penalty = penalty - 1/2*e_u*b_v*d_v';
123
124 % Continuity of u (symmetrizing term)
125 closure = closure + 1/2*d_u*b_u*e_u';
126 penalty = penalty - 1/2*d_u*b_u*e_v';
127
128 % Continuity of u (symmetric term)
129 tau = 1/4*(b_u/gamm_u + b_v/gamm_v)*tuning;
130 closure = closure - e_u*tau*e_u';
131 penalty = penalty + e_u*tau*e_v';
132
133 % Multiply by Hi and a.
134 closure = obj.Hi*obj.a*closure;
135 penalty = obj.Hi*obj.a*penalty;
136
137 end
138
139 % Returns the boundary operator op for the boundary specified by the string boundary.
140 % op -- string
141 % boundary -- string
142 function o = getBoundaryOperator(obj, op, boundary)
143 assertIsMember(op, {'e', 'd'})
144 assertIsMember(boundary, {'l', 'r'})
145
146 o = obj.([op, '_', boundary]);
147 end
148
149 % Returns square boundary quadrature matrix, of dimension
150 % corresponding to the number of boundary points
151 %
152 % boundary -- string
153 % Note: for 1d diffOps, the boundary quadrature is the scalar 1.
154 function H_b = getBoundaryQuadrature(obj, boundary)
155 assertIsMember(boundary, {'l', 'r'})
156
157 H_b = 1;
158 end
159
160 % Returns the boundary sign. The right boundary is considered the positive boundary
161 % boundary -- string
162 function s = getBoundarySign(obj, boundary)
163 assertIsMember(boundary, {'l', 'r'})
164
165 switch boundary
166 case {'r'}
167 s = 1;
168 case {'l'}
169 s = -1;
170 end
171 end
172
173 function N = size(obj)
174 N = obj.grid.size();
175 end
176
177 end
178 end