Lecture 34 - Ordinary Differential Equations - BVPSystems of Ordinary Differential Equations - BVPShooting Method for Nonlinear ODE-BVPsNonlinear Shooting Based on Secant MethodMATLAB Example in Nonlinear Shooting MethodSlide 6Nonlinear Shooting Based on Newton’s MethodSlide 8Nonlinear Shooting with Newton’s MethodSlide 10Finite-Difference MethodsFinite-Difference MethodSlide 13Slide 14Finite-Difference Method for Nonlinear BVPsSlide 16Example 14.12 - MATLABChapter 15Classification of PDEsHeat Equation: Parabolic PDESlide 21Slide 22Heat EquationExplicit MethodExplicit Euler MethodSlide 26Example: Explicit Euler MethodExampleSlide 29Slide 30Numerical StabilitySlide 32Slide 33Implicit MethodImplicit Euler MethodSlide 36Example: Implicit Euler MethodSlide 38Slide 39Slide 40Crank-Nicolson MethodGeneral Two-Level MethodExample: Crank-Nicolson MethodSlide 44Slide 45Slide 46Heat Equation with Insulated BoundaryInsulated BoundaryLecture 34 - Ordinary Differential Lecture 34 - Ordinary Differential Equations - BVP Equations - BVP CVEN 302November 28, 2001Systems of Ordinary Differential Systems of Ordinary Differential Equations - BVPEquations - BVP•Shooting Method for Nonlinear BVP•Finite Difference Method•Partial Differential EquationsShooting Method for Nonlinear Shooting Method for Nonlinear ODE-BVPsODE-BVPs•Nonlinear ODE•Consider with guessed slope t•Use the difference between u(b) and yb to adjust u’(a)•m(t) = u(b, t) - yb is a function of the guessed value t•Use secant method or Newton method to find the correct t value with m(t) = 0bay)b(y ,y)a(ybxa , )y,y,x(fyt)a(u ,yu(a) )u,u,x(fuaNonlinear Shooting Based on Secant Nonlinear Shooting Based on Secant MethodMethod•Nonlinear ODE0))b(y),b(yh( ,y)a(ybxa , )y,y,x(fyatol1)-t(i-t(i) until iterate :4 Step1)m(i2)m(i1)m(i2)t(i1)t(i1)t(it(i) estimate new a obtain to methodsecant use :3 Stepm(2)Error t(2)(a)u ,yu(a) use:2 Stepm(1)Error t(1)(a)u ,yu(a) use:1 StepaaMATLAB Example in Nonlinear Shooting MATLAB Example in Nonlinear Shooting MethodMethod•Nonlinear shooting with secant method•Convert to two first-order ODE-IVPs•Update t using the secant method )1x/(1y solution exact025.0)1(y)1y())1(y),1(y( , h1)0y(1x0 , yy2yt)0(z ,zz2z1)0(z , zzyz ,yz let221212121Nonlinear Shooting - Secant MethodNonlinear Shooting - Secant Methody(x)y’(x)Nonlinear ShootingNonlinear Shooting Based on Newton’s Method Based on Newton’s Method•Nonlinear ODECheck for convergence of m(t)bay)b(y ,y)a(ybxa , )y,y,x(fy 1(a)v 0,v(a) t(a)u ,yu(a) )uu,(x,fv )uu,(x,fvv )uu,f(x,u kauu)t,b(vmtt ,Otherwisestop ,tolm ify)tu(b,)tm( kk1kbkkNonlinear ShootingNonlinear Shooting Based on Newton’s Method Based on Newton’s Method•Nonlinear ODE-IVP•Chain Rulebay)t,b(u)t(mt)a(u ,y)a(u , )u,u,x(futuuftuuf txxft)u,u,x(f tu 1t/u(a)v' t)t,a(uvfvfvtuuftuuftutuv ,tuv then ,tuv let uux and t are independent0Nonlinear Shooting with Newton’s MethodNonlinear Shooting with Newton’s Method•Solve ODE-IVP•Construct the auxiliary equations1)0(u, 1)0u( , u)u(u2)1(, y1)0y( , y)y(y221)0(v ,0)0(vvuu2vu)u(v uu2uf)u,u,x(fu)u(uf)u,u,x(f22u22uNonlinear Shooting with Newton’s MethodNonlinear Shooting with Newton’s Method•Calculate m(t) -- deviation from the exact BC•Update t by Newton’s method),()(),()(tbutmytbutmttb),( ;)()(tbutm(t)m vtmtmtttt1it1i1iiFinite-Difference MethodsFinite-Difference Methods•Divide the interval of interest into subintervals•Replace the derivatives by appropriate finite-difference approximations in Chapter 11•Solve the system of algebraic equations by methods in Chapters 3 and 4•For nonlinear ODEs, methods in Chapter 5 may be usedi1in0xxhabh ,bx ,ax Finite-Difference MethodFinite-Difference Method•General Two-Point BVPs•Replace the derivatives by appropriate finite-difference approximationsy(b) ,y(a)bxa ),x(ry)x(qy)x(pyh2yy)x(y ,hyy2y)x(y1i1ii21ii1iixixi-1xi+1hh h hFinite-Difference MethodFinite-Difference Method•Central difference approximations•Tridiagonal system i21iiii21iiiii1i1ii21ii1irhy p2h1y qh2y p2h1ryqh2yyphyy2y21ii1ii1i1iihyy2y)x(yh2yy)x(yFinite-Difference MethodFinite-Difference Method•Central Difference ==> Tridiagonal system)/2ph1(rhrhrh )/2ph1(rhy)/2ph1(rhrhrhy)/2ph1(rh yyyy)qh2(0000)qh2(p(h/2)100p(h/2)1)qh2(p(h/2)1000)qh2( 1-n1n23222112n1-n1n2322201121n3211n2323222212Finite-Difference Method for Finite-Difference Method for Nonlinear BVPsNonlinear BVPs•Nonlinear ODE-BVPs•Evaluate fi by appropriate finite-difference approximationsy(b) ,y(a)bxa , )y,y,x(fy0fhyy2yi21ii1ixixi-1xi+1hh h hFinite-Difference Method for Finite-Difference Method for Nonlinear BVPsNonlinear BVPs•SOR method•Iterative solution•Convergence criterion)fhyy 2y()1(21yi21ii1iiP)y,y,x(fQ)'y,y,x(fQ0 withP/2h2/Qhyy2Example 14.12 -