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CSUN ME 501B - Additional Topics in Numerical Solutions of Parabolic Equations

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More numerical diffusion topics March 23 2009 Outline Additional Topics in Numerical Solutions of Parabolic Equations Complete discussion from last class Fully implicit and DuFort Frankel methods Review numerical diffusion solutions Larry Caretto Mechanical Engineering 501AB Explicit and Crank Nicholson methods Thomas algorithm Seminar in Engineering Analysis General properties of numerical algorithms for PDEs Numerical diffusion equation solutions in two or more space dimensions March 23 2009 2 Fully Implicit Method Fully Implicit BTCS Results Same as CN inputs 1 L 1 x 0 01 t 0 0005 f t x 2 5 Discretize diffusion equation at tn 1 n 1 u t i n 1 u t i u n 1 i u O t and t n i n 1 u x 2 i 2 u n 1 i 1 n 1 i 1 u 2u x 2 n 1 i O x 2 n 1 uin 1 uin u n 1 uin 11 2uin 1 i 1 O t x 2 0 t x 2 x i fuin 11 1 2 f uin 1 fuin 11 uin 2T 2 Tridiagonal system of equations Almost same work as CN and no spurious oscillations but less accuracy i 0 i 1 i 2 i 3 i 4 x 0 x 01 x 02 x 03 x 04 1000 t 0 1000 1000 1000 1000 n 0 t 0 0 1000 1000 1000 1000 n 1 t 0 0005 0 358 26 588 17 735 71 830 39 n 2 t 0 001 0 218 22 408 43 562 69 682 35 n 3 t 0 0015 0 166 26 322 13 460 74 578 96 n 4 t 0 002 0 139 05 272 65 396 35 507 18 n 5 t 0 0025 0 121 84 240 25 352 17 455 26 n 6 t 0 003 0 109 75 217 08 319 77 415 99 u0 u10 0 for all times 3 4 Fully Implicit Results II Fully Implicit Results III i 0 i 1 i 2 i 3 i 4 i 0 i 1 i 2 i 3 i 4 x 0 x 01 x 02 x 03 x 04 x 0 x 01 x 02 x 03 x 04 n 7 t 0 0035 0 100 65 199 49 294 81 385 13 n 18 t 0 009 0 60 70 121 03 180 64 239 17 n 8 t 0 004 0 93 50 185 57 274 85 360 14 n 19 t 0 0095 0 59 02 117 70 175 71 232 74 n 9 t 0 0045 0 87 68 174 19 258 43 339 38 n 20 t 0 01 0 57 47 114 62 171 16 226 79 n 10 t 0 005 0 82 82 164 67 244 62 321 81 n 21 t 0 0105 0 56 03 111 78 166 95 221 28 n 11 t 0 0055 0 78 69 156 56 232 81 306 69 n 22 t 0 011 0 54 70 109 13 163 04 216 16 n 12 t 0 006 0 75 13 149 54 222 55 293 50 n 23 t 0 0115 0 53 46 106 67 159 38 211 37 n 13 t 0 0065 0 72 00 143 38 213 53 281 87 n 24 t 0 012 0 52 30 104 36 155 96 206 88 n 14 t 0 007 0 69 24 137 93 205 52 271 52 n 25 t 0 0125 0 51 21 102 20 152 76 202 67 n 15 t 0 0075 0 66 77 133 05 198 35 262 22 Exact t 0 0125 0 50 43 100 66 150 48 199 72 n 16 t 0 008 0 64 55 128 66 191 88 253 82 Error t 0 0125 0 0 779 1 542 2 273 2 956 n 17 t 0 0085 0 62 54 124 67 186 01 246 17 5 ME 501B Engineering Analysis 6 1 More numerical diffusion topics March 23 2009 Compare Crank Nicholson Richardson Leapfrog i 0 i 1 i 2 i 3 i 4 x 0 x 01 x 02 x 03 x 04 n 18 t 0 009 0 60 65 117 177 71 234 21 n 19 t 0 0095 0 56 86 116 5 171 59 228 43 n 20 t 0 01 0 57 1 111 53 168 52 222 53 n 21 t 0 0105 0 54 43 110 47 163 53 217 57 n 22 t 0 011 0 54 19 106 68 160 64 212 45 n 23 t 0 0115 0 52 22 105 35 156 49 208 11 n 24 t 0 012 0 51 73 102 36 153 78 203 64 n 25 t 0 0125 0 50 21 100 93 150 27 199 78 Exact t 0 0125 0 50 43 100 66 150 48 199 72 Error t 0 0125 0 0 216 0 272 0 212 0 061 Use two time step central differences u u n 1 uin 1 2u i O t 2 2 t i 2 t x n n i uin 1 uin 1 2uin x 2 O x 2 Result is explicit with second order accuracy in time uin 1 uin 1 2 t x 2 u n n n i 1 ui 1 2ui u n 1 2f i u n n n i 1 ui 1 2ui However result is unstable for any f and cannot be used 7 8 DuFort Frankel DuFort Frankel Modification of Richardson method to provide stability Replace 2uin in second derivative by average at time steps n 1 and n 1 Introduces another O t 2 error u u n 1 uin 1 2u i O t 2 2 t i 2 t x n 2uin n uin 1 uin 1 2uin x 2 i Rearrange and introduce f t x 2 uin 1 uin 1 2 t x 2 u n n n 1 uin 1 i 1 ui 1 ui 2 f u n n n 1 uin 1 i 1 ui 1 ui O x uin 1 uin 1 u n T n u n 1 uin 1 t 2 i 1 i 1 2i O x 2 t 2 2 t x x 2 Result is explicit for values at time n 1 Explicit start required to get first set of values at time n 1 Can show that this is unconditionally stable 9 Truncation Error un 1i 1 uin 1 un 1i 1 uin uni 1 uni 1 Derivation in appendix to notes on numerical solutions for PDEs Set f2 1 12 f 0 288675 for minimum truncation error 2f k 2 t 2k 1 2k T 2k t n 2 2k i t k 1 k T f k 1 2k t k k 3 Old time fuin 11 1 2 f uin 1 fuin 11 uin Fully implicit n New time uin 1 f uin 1 uin 1 1 2 f uin Explicit fuin 11 2 1 f uin 1 fuin 11 CrankNicholson f u n u …


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CSUN ME 501B - Additional Topics in Numerical Solutions of Parabolic Equations

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