Richardson Extrapolation-Based Verification Method of Scientific Calculation Program without the Oracles: A Case Study

<table class="fixed-width table-group" id="tab5"><tr><td><table class="table"><colgroup><col style="width:7.64em"/><col style="width:6.13em"/><col style="width:6.14em"/><col style="width:6.13em"/><col style="width:6.12em"/><col style="width:6.12em"/><col style="width:6.15em"/></colgroup><tr><td class="thead-hr" colspan="7"><hr/></td></tr><tr class="thead"><td class="align_left">Mesh</td><td class="align_center"><span style="width: 11.714ptpx;"><svg height="12.4698pt" id="M144" style="vertical-align:-3.18147pt" version="1.1" viewbox="-0.0498162 -9.28833 11.714 12.4698" width="11.714pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M499 88L487 113C459 86 422 65 413 65C403 65 403 74 409 98L461 318C487 426 460 448 431 448C408 448 383 441 352 424C305 398 223 344 157 257H155L243 674C249 702 249 712 240 712C227 712 170 680 87 673L82 648H117C155 648 162 644 150 590L23 -4L30 -12C55 -4 81 3 105 8C113 53 131 150 143 187C199 281 323 387 372 387C390 387 393 369 380 311L328 86C313 19 319 -12 347 -12C379 -12 442 24 499 88Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.643,3.132)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"></path></g></svg></span></td><td class="align_center"><span style="width: 11.714ptpx;"><svg height="12.4698pt" id="M145" style="vertical-align:-3.18147pt" version="1.1" viewbox="-0.0498162 -9.28833 11.714 12.4698" width="11.714pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M499 88L487 113C459 86 422 65 413 65C403 65 403 74 409 98L461 318C487 426 460 448 431 448C408 448 383 441 352 424C305 398 223 344 157 257H155L243 674C249 702 249 712 240 712C227 712 170 680 87 673L82 648H117C155 648 162 644 150 590L23 -4L30 -12C55 -4 81 3 105 8C113 53 131 150 143 187C199 281 323 387 372 387C390 387 393 369 380 311L328 86C313 19 319 -12 347 -12C379 -12 442 24 499 88Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.643,3.132)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"></path></g></svg></span></td><td class="align_center"><span style="width: 11.714ptpx;"><svg height="12.5794pt" id="M146" style="vertical-align:-3.29107pt" version="1.1" viewbox="-0.0498162 -9.28833 11.714 12.5794" width="11.714pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M499 88L487 113C459 86 422 65 413 65C403 65 403 74 409 98L461 318C487 426 460 448 431 448C408 448 383 441 352 424C305 398 223 344 157 257H155L243 674C249 702 249 712 240 712C227 712 170 680 87 673L82 648H117C155 648 162 644 150 590L23 -4L30 -12C55 -4 81 3 105 8C113 53 131 150 143 187C199 281 323 387 372 387C390 387 393 369 380 311L328 86C313 19 319 -12 347 -12C379 -12 442 24 499 88Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.643,3.132)"><path d="M290 377C321 398 342 415 358 430C378 450 389 473 389 502C389 578 329 635 238 635H237C184 635 137 610 109 578L64 515L88 493C112 529 154 573 208 573S303 542 303 482C303 409 233 370 141 341L149 308C165 313 190 319 215 319C272 319 341 283 341 193C342 98 292 43 222 43C163 43 122 72 96 94C88 101 79 100 70 94C61 87 47 73 46 60C44 47 48 37 62 23C76 10 118 -12 165 -12C238 -12 430 62 430 223C430 297 379 359 290 375V377Z"></path></g></svg></span></td><td class="align_center"><span style="width: 11.714ptpx;"><svg height="12.4698pt" id="M147" style="vertical-align:-3.18147pt" version="1.1" viewbox="-0.0498162 -9.28833 11.714 12.4698" width="11.714pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M499 88L487 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422 65 413 65C403 65 403 74 409 98L461 318C487 426 460 448 431 448C408 448 383 441 352 424C305 398 223 344 157 257H155L243 674C249 702 249 712 240 712C227 712 170 680 87 673L82 648H117C155 648 162 644 150 590L23 -4L30 -12C55 -4 81 3 105 8C113 53 131 150 143 187C199 281 323 387 372 387C390 387 393 369 380 311L328 86C313 19 319 -12 347 -12C379 -12 442 24 499 88Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.643,3.132)"><path d="M158 548H390L417 615L410 623H122L83 318C105 326 143 337 185 337C296 337 350 275 350 188C350 116 308 42 225 42C164 42 122 74 100 93C90 101 82 99 72 92C60 82 51 68 50 59C48 46 52 38 66 24C82 9 125 -12 172 -12C225 -11 292 15 346 59C408 108 437 166 437 226C437 309 371 397 242 397C214 397 170 382 133 369L158 548Z"></path></g></svg></span></td><td class="align_center"><span style="width: 11.714ptpx;"><svg height="12.5794pt" id="M149" style="vertical-align:-3.29107pt" version="1.1" viewbox="-0.0498162 -9.28833 11.714 12.5794" width="11.714pt" 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colspan="7"><hr/></td></tr><tr><td class="align_left">Element</td><td class="align_center">18</td><td class="align_center">36</td><td class="align_center">72</td><td class="align_center">144</td><td class="align_center">288</td><td class="align_center">576</td></tr><tr><td class="align_left" colspan="7"><hr/></td></tr><tr><td class="align_left">Max-eigenvalue</td><td class="align_center">9.86988E-01</td><td class="align_center">9.86954E-01</td><td class="align_center">9.86945E-01</td><td class="align_center">9.86943E-01</td><td class="align_center">9.86943E-01</td><td class="align_center">9.86942E-01</td></tr><tr class="table-tr"><td colspan="7"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

<div>The calculation results of Max-eigenvalue in different refined grids.</div>

Mathematical Problems in Engineering

tab5

Table 5

Table 5: Richardson Extrapolation-Based Verification Method of Scientific Calculation Program without the Oracles: A Case Study 