Mercurial > repos > public > sbplib
annotate +scheme/Wave2d.m @ 1198:2924b3a9b921 feature/d2_compatible
Add OpSet for fully compatible D2Variable, created from regular D2Variable by replacing d1 by first row of D1. Formal reduction by one order of accuracy at the boundary point.
| author | Martin Almquist <malmquist@stanford.edu> |
|---|---|
| date | Fri, 16 Aug 2019 14:30:28 -0700 |
| parents | 5afc774fb7c4 |
| children |
| rev | line source |
|---|---|
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1 classdef Wave2d < scheme.Scheme |
| 0 | 2 properties |
| 3 m % Number of points in each direction, possibly a vector | |
| 4 h % Grid spacing | |
| 5 x,y % Grid | |
| 6 X,Y % Values of x and y for each grid point | |
| 7 order % Order accuracy for the approximation | |
| 8 | |
| 9 D % non-stabalized scheme operator | |
| 10 M % Derivative norm | |
| 11 alpha | |
| 12 | |
| 13 H % Discrete norm | |
| 14 Hi | |
| 15 H_x, H_y % Norms in the x and y directions | |
| 16 Hx,Hy % Kroneckerd norms. 1'*Hx*v corresponds to integration in the x dir. | |
| 17 Hi_x, Hi_y | |
| 18 Hix, Hiy | |
| 19 e_w, e_e, e_s, e_n | |
| 20 d1_w, d1_e, d1_s, d1_n | |
| 21 gamm_x, gamm_y | |
| 22 end | |
| 23 | |
| 24 methods | |
|
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25 function obj = Wave2d(m,lim,order,alpha) |
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26 default_arg('alpha',1); |
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27 |
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28 xlim = lim{1}; |
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29 ylim = lim{2}; |
| 0 | 30 |
| 31 if length(m) == 1 | |
| 32 m = [m m]; | |
| 33 end | |
| 34 | |
| 35 m_x = m(1); | |
| 36 m_y = m(2); | |
| 37 | |
| 38 [x, h_x] = util.get_grid(xlim{:},m_x); | |
| 39 [y, h_y] = util.get_grid(ylim{:},m_y); | |
| 40 | |
| 41 ops_x = sbp.Ordinary(m_x,h_x,order); | |
| 42 ops_y = sbp.Ordinary(m_y,h_y,order); | |
| 43 | |
| 44 I_x = speye(m_x); | |
| 45 I_y = speye(m_y); | |
| 46 | |
| 47 D2_x = sparse(ops_x.derivatives.D2); | |
| 48 H_x = sparse(ops_x.norms.H); | |
| 49 Hi_x = sparse(ops_x.norms.HI); | |
| 50 M_x = sparse(ops_x.norms.M); | |
| 51 e_l_x = sparse(ops_x.boundary.e_1); | |
| 52 e_r_x = sparse(ops_x.boundary.e_m); | |
| 53 d1_l_x = sparse(ops_x.boundary.S_1); | |
| 54 d1_r_x = sparse(ops_x.boundary.S_m); | |
| 55 | |
| 56 D2_y = sparse(ops_y.derivatives.D2); | |
| 57 H_y = sparse(ops_y.norms.H); | |
| 58 Hi_y = sparse(ops_y.norms.HI); | |
| 59 M_y = sparse(ops_y.norms.M); | |
| 60 e_l_y = sparse(ops_y.boundary.e_1); | |
| 61 e_r_y = sparse(ops_y.boundary.e_m); | |
| 62 d1_l_y = sparse(ops_y.boundary.S_1); | |
| 63 d1_r_y = sparse(ops_y.boundary.S_m); | |
| 64 | |
| 65 D2 = kr(D2_x, I_y) + kr(I_x, D2_y); | |
| 66 obj.H = kr(H_x,H_y); | |
| 67 obj.Hx = kr(H_x,I_y); | |
| 68 obj.Hy = kr(I_x,H_y); | |
| 69 obj.Hix = kr(Hi_x,I_y); | |
| 70 obj.Hiy = kr(I_x,Hi_y); | |
| 71 obj.Hi = kr(Hi_x,Hi_y); | |
| 72 obj.M = kr(M_x,H_y)+kr(H_x,M_y); | |
| 73 obj.e_w = kr(e_l_x,I_y); | |
| 74 obj.e_e = kr(e_r_x,I_y); | |
| 75 obj.e_s = kr(I_x,e_l_y); | |
| 76 obj.e_n = kr(I_x,e_r_y); | |
| 77 obj.d1_w = kr(d1_l_x,I_y); | |
| 78 obj.d1_e = kr(d1_r_x,I_y); | |
| 79 obj.d1_s = kr(I_x,d1_l_y); | |
| 80 obj.d1_n = kr(I_x,d1_r_y); | |
| 81 | |
| 82 obj.m = m; | |
| 83 obj.h = [h_x h_y]; | |
| 84 obj.order = order; | |
| 85 | |
| 86 obj.alpha = alpha; | |
| 87 obj.D = alpha*D2; | |
| 88 obj.x = x; | |
| 89 obj.y = y; | |
| 90 obj.X = kr(x,ones(m_y,1)); | |
| 91 obj.Y = kr(ones(m_x,1),y); | |
| 92 | |
| 93 obj.gamm_x = h_x*ops_x.borrowing.M.S; | |
| 94 obj.gamm_y = h_y*ops_y.borrowing.M.S; | |
| 95 end | |
| 96 | |
| 97 | |
| 98 % Closure functions return the opertors applied to the own doamin to close the boundary | |
| 99 % Penalty functions return the opertors to force the solution. In the case of an interface it returns the operator applied to the other doamin. | |
| 100 % boundary is a string specifying the boundary e.g. 'l','r' or 'e','w','n','s'. | |
| 101 % type is a string specifying the type of boundary condition if there are several. | |
| 102 % data is a function returning the data that should be applied at the boundary. | |
| 103 % neighbour_scheme is an instance of Scheme that should be interfaced to. | |
| 104 % neighbour_boundary is a string specifying which boundary to interface to. | |
| 105 function [closure, penalty] = boundary_condition(obj,boundary,type,data) | |
| 106 default_arg('type','neumann'); | |
| 107 default_arg('data',0); | |
| 108 | |
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109 [e, d] = obj.getBoundaryOperator({'e', 'd'}, boundary); |
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110 gamm = obj.getBoundaryBorrowing(boundary); |
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111 s = obj.getBoundarySign(boundary); |
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112 halfnorm_inv = obj.getHalfnormInv(boundary); |
| 0 | 113 |
| 114 switch type | |
| 115 % Dirichlet boundary condition | |
| 116 case {'D','d','dirichlet'} | |
| 117 alpha = obj.alpha; | |
| 118 | |
| 119 % tau1 < -alpha^2/gamma | |
| 120 tuning = 1.1; | |
| 121 tau1 = -tuning*alpha/gamm; | |
| 122 tau2 = s*alpha; | |
| 123 | |
| 124 p = tau1*e + tau2*d; | |
| 125 | |
| 126 closure = halfnorm_inv*p*e'; | |
| 127 | |
| 128 pp = halfnorm_inv*p; | |
| 129 switch class(data) | |
| 130 case 'double' | |
| 131 penalty = pp*data; | |
| 132 case 'function_handle' | |
| 133 penalty = @(t)pp*data(t); | |
| 134 otherwise | |
| 135 error('Wierd data argument!') | |
| 136 end | |
| 137 | |
| 138 | |
| 139 % Neumann boundary condition | |
| 140 case {'N','n','neumann'} | |
| 141 alpha = obj.alpha; | |
| 142 tau1 = -s*alpha; | |
| 143 tau2 = 0; | |
| 144 tau = tau1*e + tau2*d; | |
| 145 | |
| 146 closure = halfnorm_inv*tau*d'; | |
| 147 | |
| 148 pp = halfnorm_inv*tau; | |
| 149 switch class(data) | |
| 150 case 'double' | |
| 151 penalty = pp*data; | |
| 152 case 'function_handle' | |
| 153 penalty = @(t)pp*data(t); | |
| 154 otherwise | |
| 155 error('Wierd data argument!') | |
| 156 end | |
| 157 | |
| 158 % Unknown, boundary condition | |
| 159 otherwise | |
| 160 error('No such boundary condition: type = %s',type); | |
| 161 end | |
| 162 end | |
| 163 | |
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164 function [closure, penalty] = interface(obj, boundary, neighbour_scheme, neighbour_boundary, type) |
| 0 | 165 % u denotes the solution in the own domain |
| 166 % v denotes the solution in the neighbour domain | |
| 167 [e_u,d_u,s_u,gamm_u, halfnorm_inv] = obj.get_boundary_ops(boundary); | |
| 168 [e_v,d_v,s_v,gamm_v] = neighbour_scheme.get_boundary_ops(neighbour_boundary); | |
| 169 | |
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170 [e_u, d_u] = obj.getBoundaryOperator({'e', 'd'}, boundary); |
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171 gamm_u = obj.getBoundaryBorrowing(boundary); |
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172 s_u = obj.getBoundarySign(boundary); |
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173 halfnorm_inv = obj.getHalfnormInv(boundary); |
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174 |
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175 [e_v, d_v] = neighbour_scheme.getBoundaryOperator({'e', 'd'}, neighbour_boundary); |
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176 gamm_v = neighbour_scheme.getBoundaryBorrowing(neighbour_boundary); |
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177 s_v = neighbour_scheme.getBoundarySign(neighbour_boundary); |
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178 |
| 0 | 179 tuning = 1.1; |
| 180 | |
| 181 alpha_u = obj.alpha; | |
| 182 alpha_v = neighbour_scheme.alpha; | |
| 183 | |
| 184 % tau1 < -(alpha_u/gamm_u + alpha_v/gamm_v) | |
| 185 | |
| 186 tau1 = -(alpha_u/gamm_u + alpha_v/gamm_v) * tuning; | |
| 187 tau2 = s_u*1/2*alpha_u; | |
| 188 sig1 = s_u*(-1/2); | |
| 189 sig2 = 0; | |
| 190 | |
| 191 tau = tau1*e_u + tau2*d_u; | |
| 192 sig = sig1*e_u + sig2*d_u; | |
| 193 | |
| 194 closure = halfnorm_inv*( tau*e_u' + sig*alpha_u*d_u'); | |
| 195 penalty = halfnorm_inv*(-tau*e_v' - sig*alpha_v*d_v'); | |
| 196 end | |
| 197 | |
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198 |
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199 % Returns the boundary operator op for the boundary specified by the string boundary. |
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200 % op -- string or a cell array of strings |
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201 % boundary -- string |
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202 function varargout = getBoundaryOperator(obj, op, boundary) |
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203 assertIsMember(boundary, {'w', 'e', 's', 'n'}) |
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204 |
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205 if ~iscell(op) |
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206 op = {op}; |
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207 end |
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208 |
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209 for i = 1:numel(op) |
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210 switch op{i} |
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211 case 'e' |
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212 switch boundary |
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213 case 'w' |
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214 e = obj.e_w; |
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215 case 'e' |
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216 e = obj.e_e; |
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217 case 's' |
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218 e = obj.e_s; |
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219 case 'n' |
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220 e = obj.e_n; |
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221 end |
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222 varargout{i} = e; |
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223 |
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224 case 'd' |
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225 switch boundary |
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226 case 'w' |
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227 d = obj.d1_w; |
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228 case 'e' |
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229 d = obj.d1_e; |
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230 case 's' |
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231 d = obj.d1_s; |
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232 case 'n' |
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233 d = obj.d1_n; |
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234 end |
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235 varargout{i} = d; |
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236 end |
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237 end |
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238 |
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239 end |
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240 |
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241 % Returns square boundary quadrature matrix, of dimension |
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242 % corresponding to the number of boundary points |
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243 % |
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244 % boundary -- string |
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245 function H_b = getBoundaryQuadrature(obj, boundary) |
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246 assertIsMember(boundary, {'w', 'e', 's', 'n'}) |
|
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247 |
| 0 | 248 switch boundary |
| 249 case 'w' | |
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250 H_b = obj.H_y; |
| 0 | 251 case 'e' |
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252 H_b = obj.H_y; |
| 0 | 253 case 's' |
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254 H_b = obj.H_x; |
| 0 | 255 case 'n' |
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256 H_b = obj.H_x; |
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257 end |
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258 end |
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259 |
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260 % Returns borrowing constant gamma |
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261 % boundary -- string |
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262 function gamm = getBoundaryBorrowing(obj, boundary) |
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263 assertIsMember(boundary, {'w', 'e', 's', 'n'}) |
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264 |
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265 switch boundary |
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266 case {'w','e'} |
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267 gamm = obj.gamm_x; |
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268 case {'s','n'} |
| 0 | 269 gamm = obj.gamm_y; |
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270 end |
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271 end |
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272 |
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273 % Returns the boundary sign. The right boundary is considered the positive boundary |
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274 % boundary -- string |
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275 function s = getBoundarySign(obj, boundary) |
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276 assertIsMember(boundary, {'w', 'e', 's', 'n'}) |
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277 |
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278 switch boundary |
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279 case {'e','n'} |
| 0 | 280 s = 1; |
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281 case {'w','s'} |
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282 s = -1; |
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283 end |
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284 end |
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285 |
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286 % Returns the halfnorm_inv used in SATs. TODO: better notation |
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287 function Hinv = getHalfnormInv(obj, boundary) |
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288 assertIsMember(boundary, {'w', 'e', 's', 'n'}) |
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289 |
|
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290 switch boundary |
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291 case 'w' |
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292 Hinv = obj.Hix; |
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293 case 'e' |
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294 Hinv = obj.Hix; |
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295 case 's' |
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296 Hinv = obj.Hiy; |
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297 case 'n' |
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298 Hinv = obj.Hiy; |
| 0 | 299 end |
| 300 end | |
| 301 | |
| 302 function N = size(obj) | |
| 303 N = prod(obj.m); | |
| 304 end | |
| 305 | |
| 306 end | |
|
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307 end |