NumericalSimulationofWaveEquation

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Global Joral of Scece Froter Research: A Physcs ad Space Scece Volme 4 Isse 7 Verso. Year 4 Type : Doble Bld Peer Revewed Iteratoal Research Joral Pblsher: Global Jorals Ic.

Yeadl Islam Shakh & SM Sala Udd Samrat Iteratoal Uversty of Bsess Agrcltre ad Techology, Bagladesh Abstract- Wave eqato s a very mportat eqato appled mathematcs.

Therefore oe eeds to se mercal methods for solvg ths eqato.

We mplemet the mercal scheme by compter programmg for tal bodary vale problem ad verfy the qaltatve behavor of the mercal solto of the wave eqato.

1 Global Joral of Scece Froter Research: A Physcs ad Space Scece Volme 4 Isse 7 Verso. Year 4 Type : Doble Bld Peer Revewed Iteratoal Research Joral Pblsher: Global Jorals Ic. (USA Ole ISSN: & Prt ISSN: Nmercal Smlato of Wave Eqato By Md. Sadzzama, Md. Yeadl Islam Shakh & SM Sala Udd Samrat Iteratoal Uversty of Bsess Agrcltre ad Techology, Bagladesh Abstract- Wave eqato s a very mportat eqato appled mathematcs. Ths eqato s sed to smlate large destrctve waves fjord, lake, or the ocea geerated by sldes, earthqakes, sbsea volcaoes, meteortes. It has aalytcal solto bt t s tme cosmg. Therefore oe eeds to se mercal methods for solvg ths eqato. For ths we vestgate fte dfferece method ad preset eplct pwd dfferece scheme for oe dmesoal wave eqato, cetral dfferece scheme for secod order wave eqato. We mplemet the mercal scheme by compter programmg for tal bodary vale problem ad verfy the qaltatve behavor of the mercal solto of the wave eqato. Keywords: lear advecto eqato, eqato of cotty, wave eqato, cetral dfferece scheme, eplct pwd dfferece scheme. GJSFR-A Classfcato : FOR Code: 9999 NmercalSmlatoofWaveEqato Strctly as per the complace ad reglatos of : 4. Md. Sadzzama, Md. Yeadl Islam Shakh & SM Sala Udd Samrat. Ths s a research/revew paper, dstrbted der the terms of the Creatve Commos Attrbto-Nocommercal 3. Uported Lcese permttg all o commercal se, dstrbto, ad reprodcto ay medm, provded the orgal work s properly cted.

2 Nmercal Smlato of Wave Eqato Md. Sadzzama α, Md. Yeadl Islam Shakh σ & SM Sala Udd Samrat ρ Abstract- Wave eqato s a very mportat eqato appled mathematcs. Ths eqato s sed to smlate large destrctve waves fjord, lake, or the ocea geerated by sldes, earthqakes, sbsea volcaoes, meteortes. It has aalytcal solto bt t s tme cosmg. Therefore oe eeds to se mercal methods for solvg ths eqato. For ths we vestgate fte dfferece method ad preset eplct pwd dfferece scheme for oe dmesoal wave eqato, cetral dfferece scheme for secod order wave eqato. We mplemet the mercal scheme by compter programmg for tal bodary vale problem ad verfy the qaltatve behavor of the mercal solto of the wave eqato. Keywords: lear advecto eqato, eqato of cotty, wave eqato, cetral dfferece scheme, eplct pwd dfferece scheme. I. Itrodcto W ave eqato s sed to derstadg of tsams, asssts warg systems, asssts bldg of harbor protecto (break maters, recogze crtcal coastal areas, hd cost hstorcal tsam (assst geologsts. It has aalytcal solto bt t s tme cosmg. How over wth rapd developmets of mercal methods ad compter techology the system ca be solved mercally. Nmercal smlato[] s very mch challegg. May scetfc grops are volved dealg wth ths problem. The am of ths thess wll be provde a easy way to solvg wave eqato. I ths thess we se fte dfferece scheme kow as cetral dfferece scheme [3], [4], eplct pwd dfferece scheme [8],[6]. I the frst secto we trodced symbols ad otatos. I the secod secto we trodced the frst ad secod order wave eqato, method of characterstcs, D Alembert solto ad aalytcal solto of the wave eqato. I the thrd secto we dscssed mercal methods. I the last we dscssed abot or mercal epermets ad reslts. II. Mathematcal Models ad Methods a Symbols ad Notato Let Ω R d, d {, } be a rego occped by a fld flow, ad let [ t, T ] be a tme terval wth Athor α: Lectrer, Departmet of Mathematcs, Iteratoal Uversty of Bsess Agrcltre ad Techology, Dhaka, Bagladesh. e-mal: sadzzama.j@gmal.com Athor σ: Lectrer, Departmet of Mathematcs, Shekh Ltfor Rahma Adarsha Govt. college, Kotalpara, Gopalgoj, Bagladesh. e-mal: yeadl_j@yahoo.com Athor ρ: Stde Jahagr agar Uversty, Dhaka, Bagladesh. e-mal: samrat73@gmal.com t T. A arbtrary pot Ω s deoted by T X =,..., d. For the descrpto of a geeral steady compressble fld flow, we trodce the qattes: The desty ρ = ρ, the velocty vector T V = V = ( v,... vd, the pressre P = P the eergy desty E = E. We deote the desty of the eteral force by T f = f = ( f(,, f,... f d, the mass fl by q = q for the descrpto of the vscos flow, let λ ad µ deote the coeffcet of ketc vscosty respectvely. b The eqato of cotty [5] s gve by ρ +.( ρv = c Frst order wave eqato The frst -order wave (advecto eqato [6] s ( c > y + c =,, = ( d Wave eqato d Oe dmesoal wave eqato [] s Ital codto: = c (, = ( ad (, v ( t = Formally, we ca wrte Laplace eqato as: tt ( c ( = ( + c ( c e D Alembert s solto We get the characterstcs system from ( as d d Itegratg both sde of (3 the we get ( ( ( = ± c (3 Global Joral of Scece Froter Research Volme XIV Iss e VII V erso I Year Global Jorals Ic. (US

3 Nmercal Smlato of Wave Eqato Year 4 34 Global Joral of Scece Froter Research Volme XIV Iss e VII V erso I = ct + ξ (whe c s postve (4 Solvg (.4 ad (.5 the we get = ( η + ξ ad t = ( η ξ c (whe c s egatve (5 = ct +η Itrodcg ew varable η = + ct ad ξ = ct We cosder, w ( η, ξ = = [ ( η ξ, ( η + ξ ] (6 c w =. +. = + c w = ξ + = [ c ] = Itegratg (6 wth respect to ξ we ge w = f (η ξ + 4 (sg (7 Aga Itegratg (6 wth respect to η we ge Now we get from (6 ad (8 w = f ( η ξ (8 w = f ( η ξ = f ( + c c Ths s the geeral form of solto of (. We se tal codto to determe f ad g. For t= we ge (9 (, = = f ( ( f g ξ ξ t (, = v ( = [. +. ] t= Itegratg ( we ge c f ( cg ( ( f ( g( = v ( s ds + k ( c Solvg (ad(we ge f ( = ( + v( s ds + k (3 c From (9 we ge f ( = ( + v( s ds + k (4 c = { ( + c + ( c + c v ( s ds wave + ct ct Ths s called D Alemberts solto for -D eqato [7]. f Method of characterstc Cosder the IVP (t + cc (t =, < <, tt >, = ff(, < <, Where c=costat. If we measre the rate of chage of from movg posto gve by the cha rle, dd dddd t (t = tt(( t + t (t (t The frst term o the rght-had sde above s the chage at a fed pot whle the secod oe s the chage resltg from the movemet of the observato posto. Assmg that (t = cc, c from the PDE we see that dd dddd t (t = tt t (t + cc t (t = That s = costat as perceved from the movg observato pot. The posto of ths pot s obtaed by tegratg ts velocty ( = c c = = + c ( Ths formla defes a famly of les the -plae, whch are called characterstcs (Fg... As metoed above, the characterstcs have the property that ( takes a costat vale alog each oe of them (bt the tegral, dfferet costat vales o dfferet characterstcs t o(, Fgre. : Characterstc plae 4 Global Jorals Ic. (US

4 Nmercal Smlato of Wave Eqato Hece, to fd the vale of the solto at we cosder the characterstcs throgh of eqato = ct + whch tersects the as at (,.Sce s a costat o ths le, ts vale at s the same as at (,.Bt the latter s kow from the IC, so ( = f ( The parameter s ow replaced from the eqato of the characterstcs le: = ct.ths the solto of the gve IVP s = f ( c Ths formla shows that at a fed tme t the shape of the solto s the same as attt =,, bt s shfted by ct alog the as. I other words, the shape of the tal data fcto travels the postve (egatve drecto wth velocty c f c > ( c < whch meas that the solto s a wave. III. Nmercal Methods a Eplct Upwd Dfferece Scheme We ow descrbe the eplct pwd dfferece scheme for eample we take lear advecto eqato. + = Wth tal codto (, = ( ad left-had bodary codto a, = ( a (5 b Cetral dfferece scheme for Wave eqato I ths chapter we vestgate the fte dfferece scheme for the Wave eqato [7].For a gve velocty the geeral form of wave eqato s ( = c (, a, t b tt (6 Wth bodary codto, = ad a, = for t b ad tal codto s = f ( ad t (, = g( for < < a. For eqdstace grd, wth temporal step sze ad spatal step sze the dscretzato of tt ( ad ( we se the cetral dfferece formla the we get tt t +, + t, ( = + o( (7 Ad + + ( = + o(...(8 We cosder form grd that s + = + ad t + =.Drop the terms o( ad o( ad se the appromato for, t (7 ad (8. The for eqato (5 we get = ( γ = c + + γ ( + Where γ = c + + Ths s the cetral dfferece scheme for wave eqato. Stecl: + λ ( γγ IV. + γ + Fgre. : Stecl of wave eqato Nmercal Epermets ad Reslts We develop a compter program (code Matlab scetfc programmg lagage ad the mplemet the cetral dfferece scheme for wave eqato. a Isertg data We mplemeted the cetral dfferece scheme for the mercal smlato for the wave eqato. We mplemeted the scheme for artfcal tal ad bodary data ad verfy the qaltatve behavor of the mercal solto of wave eqato. The ma parts of the mplemetato of or mercal schemes are gve the followg algorthm. Ipt: t ad the mber of grd pots of tme ad space respectvely. If T the rght ed pot of [,T] b the rght ed pot of [,b] the tal velocty apply as tal codto. b the velocty at the bodary po apply as a bodary vale. Global Joral of Scece Froter Research Volme XIV Iss e VII V erso I Year Global Jorals Ic. (US

Year 4 36 Global Joral of Scece Froter Research Volme XIV Iss e VII V erso I A Nmercal Smlato of Wave Eqato Otpt: the solto matr. t Italzato: dt = the temporal grd sze. t b d = the spatal grd sze.

b Reslts To test the accracy of the mplemetato of the mercal scheme for wave eqato, we dscss or epermet ad reslts der geeratg the cases.

We choose the parameters as =.8, = 6.8, c =.5, = (,π, t = (, wth tal codto = s ad bodary codto = t ad = ad r the propagato for ts. I ths case we see the fgre plot form ad mesh form as follows.

5 Year 4 36 Global Joral of Scece Froter Research Volme XIV Iss e VII V erso I A Nmercal Smlato of Wave Eqato Otpt: the solto matr. t Italzato: dt = the temporal grd sze. t b d = the spatal grd sze. c =.5 a costat dt gm = c *. d gma = * gm * gm. Step : calclato for the scheme for jj = 3: t for = : j, = gma * j. + gm * gm * ( j, + j, j, Step : Otpt. Step 3: Step 3: Graph presetato. Stop. b Reslts To test the accracy of the mplemetato of the mercal scheme for wave eqato, we dscss or epermet ad reslts der geeratg the cases. Case : I ths case we cosdered the frst order wave eqato (lear advecto + c =. We cosdered the aalytcal solto of oe dmesoal frst order wave eqato ad performed the epermet matlab. We choose the parameters as =.8, = 6.8, c =.5, = (,π, t = (, wth tal codto = s ad bodary codto = t ad = ad r the propagato for ts. I ths case we see the fgre plot form ad mesh form as follows. Case : I ths case we cosder the frst order wave eqato(lear advecto + c = ad perform the mercal epermet for three tme step. We choose =.8, = 6.8, c =.5, = (,π, t = (, wth tal codto = s ad bodary codto = t ad = ad r the propagato for ts. We preset the solto for the three dfferet tme step as show the fgre. As epected we observer that the tal vale movg forward wth respect to tme. Fgre : Nmercal solto of frst order wave eqato plot form ad mesh form sg eplct pwd dfferece scheme Case 3: I ths case we cosder the aalytcal solto of the wave eqato = c ad perform the epermet matlab. We choose the parameters as =., = 6.8, c =.5, = (,π, t = (, wth tal codto = s ad bodary codto = t ad = ad r the propagato for ts. I ths case we see the fgre plot form ad mesh form as follows. Fgre : Aalytcal solto of frst order wave eqato plot form ad mesh form Fgre 3 : Aalytcal solto of D wave eqato plot form ad mesh form 4 Global Jorals Ic. (US

Nmercal Smlato of Wave Eqato Case 4: I ths case we perform the mercal epermet for the eqato = c ad perform the epermet matlab. We choose the parameters as =., = 6.8, c =.

Fgre 4 : Nmercal solto of D wave eqato plot form ad mesh form sg cetral dfferece scheme V. Coclso I ths paper we cosdered wave eqato whch s a fdametal partal dfferetal eqato fld mechacs.

6 Nmercal Smlato of Wave Eqato Case 4: I ths case we perform the mercal epermet for the eqato = c ad perform the epermet matlab. We choose the parameters as =., = 6.8, c =.5, = (,π, t = (, wth tal codto = s ad bodary codto = t ad =. I ths case the tal cofgrato s ot movg forward bt oly smears ot whch s typcal behavor of the wave eqato. Fgre 4 : Nmercal solto of D wave eqato plot form ad mesh form sg cetral dfferece scheme V. Coclso I ths paper we cosdered wave eqato whch s a fdametal partal dfferetal eqato fld mechacs. Frst we showed eqato of cotty, frst order wave eqato, secod order wave eqato, method of characterstcs, D Alembert solto. Fally we showed the mercal reslt of frst order wave eqato based o eplct pwd dfferece scheme ad secod order wave eqato based o cetral dfferece scheme agrees wth basc qaltatve behavor of ths eqato. Refereces Référeces Referecas. Mathematcal modelg ad mercal smlato of the cardovasclar system-alfo qartero ad Lca Formagga.. Kt Adress Le, The Wave Eqato D ad D Dept. of Iformatcs, Uversty of Oslo. 3. Joh C. Strkwedra, Fte Dfferece Schemes ad Partal Dfferetal Eqato Secod Edto. 4. Peter J. Olver, Nmercal Aalyss Lectre Note dg/teachg /BLT/sec.pdf 6. Nmercal Methods for Hyperbolc Systems-Erc Soedrcker-3 7. Partal Dfferetal Eqatos-Stepa Tersa, Aprl L. S. Adellah, Fte Dfferece Method-Eplct Upwd Dfferece Scheme Lectre ote 7. Global Joral of Scece Froter Research Volme XIV Iss e VII V erso I Year Global Jorals Ic. (US

7 Nmercal Smlato of Wave Eqato Year 4 38 Global Joral of Scece Froter Research Volme XIV Iss e VII V erso I A Ths page s tetoally left blak 4 Global Jorals Ic. (US

u(x, t) = u 0 (x ct). This Riemann invariant u is constant along characteristics λ with x = x 0 +ct (u(x, t) = u 0 (x 0 )):

u(x, t) = u 0 (x ct). This Riemann invariant u is constant along characteristics λ with x = x 0 +ct (u(x, t) = u 0 (x 0 )): x, t, h x The Frst-Order Wave Eqato The frst-order wave advecto eqato s c > 0 t + c x = 0, x, t = 0 = 0x. The solto propagates the tal data 0 to the rght wth speed c: x, t = 0 x ct. Ths Rema varat s costat