Adaptive Time-Stepping Using Control Theory for the Chemical Langevin Equation

<div>The Schlögl model: the histogram at time <svg height="9.55092pt" id="M214" style="vertical-align:-0.2129993pt" version="1.1" viewbox="-0.0498162 -9.33792 37.3698 9.55092" width="37.3698pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M620 675H597C578 656 570 650 541 650H144C112 650 104 653 94 675H72C59 618 42 552 23 493L53 491C71 534 88 564 105 585C124 608 144 615 238 615H290L197 121C182 40 174 34 88 28L82 0H361L367 28C275 34 266 38 281 121L374 615H441C522 615 543 608 553 583C562 560 566 531 565 493L597 494C603 551 612 629 620 675Z" id="g113-85"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="637" vert-adv-y="637"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,12.42,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,24.157,0)"><path d="M456 178V225H360V632H320C217 496 115 347 20 206V178H280V106C280 40 276 34 189 27V0H445V27C364 34 360 39 360 106V178H456ZM280 225H82C149 335 214 431 278 520H280V225Z" id="g113-53"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,30.661,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z" id="g113-49"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g></svg> of the species <svg height="9.04777pt" id="M215" style="vertical-align:-0.04981995pt" version="1.1" viewbox="-0.0498162 -8.99795 10.5029 9.04777" width="10.5029pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M748 650H522L515 622L546 617C580 611 587 604 565 575C518 513 469 451 419 393C376 474 349 534 330 580C318 609 325 612 361 618L383 622L392 650H151L144 622C214 616 224 612 257 543L360 327C270 218 187 124 159 95C106 40 92 34 26 28L17 0H252L259 28L236 31C189 37 188 47 209 78C249 136 308 210 377 294L478 79C494 44 487 37 449 32L418 28L409 0H673L680 28C596 34 591 39 554 116L436 361C526 469 574 521 604 553C659 612 669 614 739 622L748 650Z" id="g113-89"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="765" vert-adv-y="765"></glyph.data></g></svg> obtained with the SSA (green) and the PI-adaptive method for the CLE with tolerances <svg height="12.2911pt" id="M216" style="vertical-align:-0.2129993pt" version="1.1" viewbox="-0.0498162 -12.0781 45.2331 12.2911" width="45.2331pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M387 375C387 402 357 448 257 448C172 448 82 404 82 326C82 289 108 255 156 241V239C85 223 23 181 23 116C23 39 89 -12 182 -12C265 -12 336 31 378 91L361 114C320 73 269 47 216 47C157 47 115 82 115 137C115 191 160 219 218 219C243 219 262 218 272 217L304 259L302 266C295 265 281 264 255 264C195 264 163 294 163 335C163 377 200 416 249 416C293 416 321 389 329 342C331 332 335 329 341 329C355 329 387 352 387 375Z" id="g113-247"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="410" vert-adv-y="410"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,9.341,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,21.079,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z" id="g113-50"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" 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horiz-adv-x="487" vert-adv-y="487"></glyph.data></g></svg> (blue) and <svg height="12.2911pt" id="M217" style="vertical-align:-0.2129993pt" version="1.1" viewbox="-0.0498162 -12.0781 45.2331 12.2911" width="45.2331pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M387 375C387 402 357 448 257 448C172 448 82 404 82 326C82 289 108 255 156 241V239C85 223 23 181 23 116C23 39 89 -12 182 -12C265 -12 336 31 378 91L361 114C320 73 269 47 216 47C157 47 115 82 115 137C115 191 160 219 218 219C243 219 262 218 272 217L304 259L302 266C295 265 281 264 255 264C195 264 163 294 163 335C163 377 200 416 249 416C293 416 321 389 329 342C331 332 335 329 341 329C355 329 387 352 387 375Z" id="g113-247"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="410" vert-adv-y="410"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,9.341,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,21.079,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z" id="g113-50"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,27.582,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z" id="g113-49"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,34.085,-5.984)"><path d="M556 236V289H56V236H556Z" id="g54-33"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="611" vert-adv-y="611"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,39.88,-5.984)"><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" id="g50-52"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g></svg> (red). The simulation uses 10,000 paths.</div>

Journal of Applied Mathematics

fig3

Figure 3

Figure 3: Adaptive Time-Stepping Using Control Theory for the Chemical Langevin Equation 