有限差分法解偏微分方程式

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有限差分法解偏微分方程式第一節偏微分方程已知二階偏微分方程222uuuuuABCfx,t,u,,022xxttxt2若B4AC0，則稱上式為拋物線型偏微分方程。2若B4AC0，則稱上式為雙曲線型偏微分方程。2若B4AC0，則稱上式為橢圓型偏微分方程。第二節Heatconduction偏微分方程之數值解HeatConductionEquation：2u1u，0xL，t022xt邊界條件與初始條件必須給定。L將x分成n段xh，並取t方向增量ktn【方法一】顯性近似法：考慮節點i,j處之差分公式：gridpoints：xi,tj，uijuxi,tj二階偏微分為中央差分近似2u2uuu2uuui1,ji,ji1,ji1,ji,ji1,j222xxh一階偏微分為前向差分近似uui,j1ui,jui,j1ui,jttk2u1u代入熱傳方程，得22xtu2uuuui1,ji,ji1,j1i,j1i,j22hk1 移項整理得2kuij2uijuijuijuij21,,1,,1,h2k令*代入，得節點i,j之有限差分近似公式2hui,j1ui,j*ui1,j2ui,jui1,j，i0,1,2,,n，j0,1,2,其中j0為初始條件，u為給定。i,0移項得PDEfinitedifferenceformulafortheheatequation：ui,j1*ui1,j12*ui,j*ui1,ji0及in為邊界條件，u，u為給定。0,jn,jIf0*0.5，thentheapproximationsui,jconvergetothesolutionux,t【方法二】隱性近似法：考慮節點i,j1處之差分公式(注意隱性近(i,j+1)與顯性法(i,j)不同，j+1lineisto-be-determined)二階偏微分為中央差分近似，2u2uuu2uuui1,j1i,j1i1,j1i1,j1i,j1i1,j1222xxh一階偏微分為前向差分近似uui,j1ui,jui,j1ui,jttk2u1u代入熱傳方程，得22xtu2uuuui1,j1i,j1i1,j11i,j1i,j22hk移項整理得2kuij2uijuijuijuij21,1,11,1,1,h2k令*代入，得節點i,j1之有限差分近似公式2h2 12*u*uuu，i0,1,2,,n，j0,1,2,i,j1i1,j1i1,j1i,j其中j0為初始條件，u為給定。i,0i0及in為邊界條件，u，u為給定。0,jn,j空間網格點：i0到in各節點代入展開得1.當i1，代入差分通式12*u*uuu，得i,j1i1,j1i1,j1i,j12*u*uuu1,j12,j10,j11,j移項得12*u*uu*uu*u1,j12,j11,j0,j11,j02.當in1，代入差分通式12*u*uuu，得i,j1i1,j1i1,j1i,j12*u*uuun1,j1n,j1n2,j1n1,j移項得12*u*uu*uu*un1,j1n2,j1n1,jn,j1n1,jn最後代入整理得矩陣形式如下：12**00u1,j1u1,j*u0*12**0u2,j1u2,j0*12*0u3,j1u3,j00012*un1,j1un1,j*un【觀念】※隱性法為無條件穩定※顯性法為條件穩定（*0.5），時間網格距愈小愈好。3 第三節雙曲線型偏微分方程WaveEquation：22u1u，0xL，t0222xct邊界條件與初始條件必須給定。L將x分成n段xh，並取t方向增量ktn考慮節點i,j處之差分公式二階偏微分為中央差分近似2u2uuu2uuui1,ji,ji1,ji1,ji,ji1,j222xxh及2u2uuu2uuui,j1i,ji,j1i,j1i,ji,j1222ttk22u1u代入波動方程，得222xctu2uu1u2uui1,ji,ji1,ji,j1i,ji,j1222hck移項整理得22cku1,2u,u1,u,12u,u,12ijijijijijijh222ck令代入，得節點i,j之有限差分近似公式2h2uijuij2uijui1,j2ui,jui1,j,1,1,或22uuu21u,u,1，i0,1,2,,n，j0,1,2,i,j1i1,ji1,jijiju其中j0為初始條件，ux,0uU及x,0V為給定。代入得ii,0iiit22uuu21uui,1i1,0i1,0i,0i,14 u上式中有u項，此項計算可由初始條件tV，利用中央差分近似一階微i,10it分值，即uui,1ui,1ui,1ui,1tV0it2t2k移項得u2kVui,1ii,1i0及in為邊界條件，u，u為給定。0,jn,j代回原差分近似式得22uUU21U2kVui,1i1i1iii,1移項得222uUU21U2kVi,1i1i1ii或22uUU1UkVi,1i1i1ii2第四節橢圓型偏微分方程LapaceEquation：222uuu0，0xa，0yb22xyPoisson’sEquation：222uuufx,y，0xa，0yb，fx,y為給定之sourcefunction。22xyHelmholtz’sEquation：222uuugx,yufx,y，0xa，0yb22xy5 L將x分成n段xh，並取y方向增量kyn考慮節點i,j處之差分公式二階偏微分為中央差分近似2u2uuu2uuui1,ji,ji1,ji1,ji,ji1,j222xxh及2u2uuu2uuui,j1i,ji,j1i,j1i,ji,j1222yyk22uu代入Laplace方程0，得22xyu2uuu2uu2i1,ji,ji1,ji,j1i,ji,j1ui,j22hk移項整理得u2uuu2uu2i1,ji,ji1,ji,j1i,ji,j1ui,j22hkk令r代入，得節點i,j之有限差分近似公式h222ru2ruruu2uu2i1,ji,ji1,ji,j1i,ji,j1ui,j2222rhrh或222ruruuu21ru2i1,ji1,ji,j1i,j1i,jui,j22rhi0,1,2,,n，j0,1,2,,n當xy及r1，代入得uuuu4u2i1,ji1,ji,j1i,j1i,jui,j2huuuui1,ji1,ji,j1i,j1Therefore,ui,j4The“relaxation”methodcanbeusedtofindtheresults.6 BoundaryConditionsDirichlet:Thevaluesofonlythefunction,u,arespecifiedonaboundary.Neumann:Thevaluesofonlythenormalderivativesofthefunctionaregivesonaboundary.Cauchy:Thevaluesofboththefunctionanditsnormalderivativearespecifiedonthesameboundary.Finitedifferenceexpressions:7