Outline. Review Small Grid (N = 6, M = 5) Review Finite Differences. Review N = 6, M = 5 Matrix. More numerical elliptic PDEs March 30, 2009
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3 ore nmercl ellptc Ds rch, 9 Centrl Derence pproch Generl condton /s b c b centrl derences rte enerl eqton or bondr nodes tht contns cttos node otsde reon rte centrl derence eqton or bondr condton olve ths eqton or potentl t cttos node nd se reslt to replce ths vle n enerl eqton t bondr Inclde bondr nodes n tertons orth ondr mple Fnte-derence eqton t Fcttos rom centrl-derence bondr condton eqton t b c c b orth ondr mple II Combne eqtons to elmnte c b b oded coecents t terted bondres: new,, nd ; ; nd mltpled b c 5 CD Generl ondr oded est bondr coecents b c oded north bondr coecents b c 6 CD Generl ondr II oded west bondr coecents b c 7 oded soth bondr coecents b c Forwrd/ckwrd Derence Generl condton, /s b c Use orwrd derences t or nd bckwrd derences t or btn eqton or bondr potentl n terms o two nodes n rom bondr Combne ths eqton wth enerl eqton or rst node n rom the bondr to elmnte nknown bondr potentl o terton on bondr vles, whch re ond t end o tertons Dervton t end o ths presentton 8 5 nneern nlss
4 ore nmercl ellptc Ds rch, 9 5 nneern nlss 9 est nd oth ondr b c b b b c b c b b b c nchned nd nchned nd orth nd st ondr b c b b b c b c b b b c nchned nd nchned nd Compct ppromtons Dene centrl-derence opertor, [ ] Compct second dervtve To ppl ths eqton, we hve to mltpl throh b denomntor nd then ppl the opertor there Fnl Reslt ch vle o s lnked to eht nerest nehbors so ech nte derence eqton hs nne terms Uses wehted vere sorce term Dervton t end o ths presentton [ ] Fnl Reslt or h For, [ ] [ ] h ondr Crosses Grd Dene new nd to dene bondr Get dervtve epressons or neven rd spcn...! '' ' '' h
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6 ore nmercl ellptc Ds rch, 9 5 nneern nlss 6 ddtonl terls ldes to present detled dervtons o the enerl bondr condtons sn one-sded derences ee Homn or other dervtons o centrl-derence bondr eqtons The dervton o the hher order derence method sn compct derences s shown on sldes to 7 Generl ondr mple Look t est bondr; other bondres ollow smlr dervton olve one-sded, second-order rst dervtve epresson or c b b c b b c Generl ondr mple II Combne bondr condton wth eqton or rst node rom bondr b c b c b b Generl ondr mple III ondr node no loner n eqton tll solve or nteror nodes onl wth solver Fnd bondr vle ter solton complete Use moded eqton coecents b c b b b c 5 Reslts or ther ondres bsttte revsed coecents or ornl coecents n nte-derence verson o D mlr pproch, wth derent eqtons, t ll bondres Iterte onl on nonbondr nodes nd nd bondr potentls ter tertons complete Dervtons ollow here 6 oth ondr Reslt smlr to est bondr b c b c b b
7 ore nmercl ellptc Ds rch, 9 5 nneern nlss 7 7 orth nd st ondres me pproch wth bckwrd derence epressons Look t st s emple c b b c b b c 8 st ondr Combne bondr condton wth eqton or lst node beore bondr b c b c b b 9 st ondr II Fnl reslts b c b c b b b c b b orth ondr Reslt or bondr smlr to b c b c b b D Compct Derences Compct derences or osson eqton Dene nd drecton derence opertors econd prtl dervtve epressons [ ] [ ] ppl to / /, osson Compct Derences ltpl b / nd se / e know nd, bt wht s
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