CIP Method of Characteristics for the Solution of Tide Wave Equations

<div>Space-time diagram that shows the characteristics <span class="nowrap"><svg height="11.0031pt" id="M51" style="vertical-align:-0.2064009pt" version="1.1" viewbox="-0.0498162 -10.7967 14.8953 11.0031" width="14.8953pt" 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="M645 631C614 643 545 666 457 666C215 666 23 519 23 283C23 90 158 -16 337 -16C412 -16 489 2 522 10C543 39 590 127 606 167L580 181C519 89 459 18 348 18C201 18 122 136 122 287C122 464 244 632 435 632C544 632 602 595 608 472L639 475C643 526 645 581 645 631Z" id="g113-68"></path></g><g transform="matrix(.0091,0,0,-0.0091,8.684,-5.741)"><path d="M556 297V350H337V548H275V350H56V297H275V85H337V297H556ZM556 -23V30H56V-23H556Z" id="g54-37"></path></g></svg>.</span> The Riemann invariants consist of <svg height="14.6307pt" id="M52" style="vertical-align:-3.834001pt" version="1.1" viewbox="-0.0498162 -10.7967 13.8409 14.6307" width="13.8409pt" 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="M485 501C480 550 475 615 475 650H42V622C119 616 127 606 127 525V126C127 44 119 34 41 28V0H311V28C220 34 212 43 212 126V584C212 612 216 615 244 615H318C382 615 404 608 420 587C435 568 445 541 455 497L485 501Z" id="g113-132"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.595,-5.741)"><path d="M556 297V350H337V548H275V350H56V297H275V85H337V297H556ZM556 -23V30H56V-23H556Z" id="g54-37"></path></g><g transform="matrix(.0091,0,0,-0.0091,5.486,3.784)"><path d="M870 650H600L593 618C681 609 684 605 669 516L641 367H261L286 516C302 603 313 610 391 618L398 650H132L123 618C214 610 218 606 200 516L127 130C113 46 107 40 25 32L18 0H290L298 32C204 40 200 42 214 130L250 321H631L596 131C580 47 573 39 484 32L475 0H753L762 32C673 39 668 46 685 130L757 516C772 606 778 610 861 618L870 650Z" id="g50-73"></path></g></svg> and <span class="nowrap"><svg height="11.0031pt" id="M53" style="vertical-align:-0.2064009pt" version="1.1" viewbox="-0.0498162 -10.7967 13.1989 11.0031" width="13.1989pt" 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="M515 96L502 119C471 88 431 62 423 62C416 62 411 70 417 101C440 223 469 341 497 448H486L412 422L380 277C330 188 210 57 152 57C137 57 126 69 139 124L195 366C210 431 205 448 182 448C155 448 89 413 23 350L36 326C73 354 103 376 112 376C118 376 118 365 113 340L61 118C54 90 52 68 52 51C52 0 75 -12 98 -12S151 -3 181 17C242 58 305 119 362 193H364L345 104C323 3 339 -12 359 -12C390 -12 464 35 515 96Z" id="g113-118"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.994,-5.741)"><path d="M556 297V350H337V548H275V350H56V297H275V85H337V297H556ZM556 -23V30H56V-23H556Z" id="g54-37"></path></g></svg>,</span> <svg height="11.0031pt" id="M54" style="vertical-align:-0.2064009pt" version="1.1" viewbox="-0.0498162 -10.7967 13.3816 11.0031" width="13.3816pt" 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="M529 97L508 118C475 75 449 58 438 58C428 58 421 66 415 104C393 234 374 403 364 496C345 670 307 712 254 712C220 712 174 691 153 669L161 645C176 653 194 658 206 658C237 658 261 640 278 562C287 522 290 483 293 434C223 269 110 105 23 9L32 -12C59 -6 85 0 108 7C152 64 251 252 300 366C307 297 315 221 337 82C346 24 363 -12 393 -12C425 -12 475 13 529 97Z" id="g113-233"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.174,-5.741)"><path d="M556 297V350H337V548H275V350H56V297H275V85H337V297H556ZM556 -23V30H56V-23H556Z" id="g54-37"></path></g></svg> are characteristic speeds.</div>

Advances in Mathematical Physics

fig2

Figure 2

Figure 2: CIP Method of Characteristics for the Solution of Tide Wave Equations 