三阶Runge-Kutta最优算法

MATHEMATICAAPPLICATA2005,18():57613Runge2Kutta,,(,066004):,,,,Runge2Kutta.:;;;Runge2Kutta:O246AMS(2000):65L06;65K10:A:100129847(2005)20057205y=f(x,y),axb,y(x0)=y0.(1)f(x,y),y=y(x).(1)y=y(x),y(xn+1)=y(xn)+hy(xn)+h22!y(xn)+h33!yÊ(xn)+,(2)y(x),dydx=f(x,y)f.yn+1=yn+vi=1iki,i,vf,kiki=hfxi+cih,yn+hi-1j=1aijkj,c1=0,i=1,2,,v.,,,.Runge2Kutta.Runge2Kuttayn+1=yn+h(1k1+2k2+3k3),k1=f(xn,yn),k2=f(xn+ph,yn+phk1),k3=f(xn+qh,yn+qh(rk1+sk2)).(3)(3),[1],3:2004203216:,,,,,:.r+s=1,(a)1+2+3=1,(b)2p+3q=12,(c)2p2+3q2=13,(d)3pqs=16.(e),,Runge2Kutta,.(i)yn+1h4.Taylor,k2,k3(xn,yn).,D=55x+fn55y,D2=525x2+2fn525x5y+f2n525y2,D3=535x3+3fn535x25y+3f2n535x5y2+f3n535y3.k2=f(xn+ph,yn+phk1)=fn+phDfn+12!p2h2D2fn+13!p3h3D3fn+,(4)k3=f(xn+qh,yn+qh(rk1+sk2))=f(xn,yn)+qh5fn5x+(rk1+sk2)5fn5y+12!q2h252fn5x2+2(rk1+sk2)52fn5x5y+(rk1+sk2)252fn5y2+13!q3h353fn5x3+3(rk1+sk2)53fn5x25y+3(rk1+sk2)253fn5x5y2+(rk1+sk2)353fn5y3+.,rk1+sk2=rfn+sfn+sphDfn+12!sp2h2D2fn+13!sp3h3D3fn+,rk1+sk2=fn+sphDfn+12!sp2h2D2fn+.(5)(a)13!sp3h3D3fn+.(5)k3k3=f(xn+qh,yn+qh(rk1+sk2))=fn+qh5fn5x+fn+sphDfn+12!sp2h2D2f+5fn5y+12!q2h252fn5x2+2fn+sphDfn+12!sp2h2D2fn+52fn5x5y+f2n+fnsphDfn+12!fnsp2h2D2fn++fnsphDfn+s2p2h2(Dfn)2852005+12!s2p3h3DfnD2fn++12!fnsp2h2D2fn+12!s2ph3h3DfnD2fn+14s2p4h4(D2fn)2++13!fnsp3h3D3fn+52fn5y2+13!q3h353fn5x3+3(fn+sphDfn+)53fn5x25y+3(f2n+)53fn5x5y2+(f3n+)53fn5y3+=f(xn,yn)+qh5fn5x+fn+sphDfn+12!sp2h2D2fn5fn5y+12!q2h252fn5x2+2(fn+sphDfn)52fn5x5y+(f2n+2fnsphDfn)52fn5y2+13!q3h353fn5x3+3fn53fn5x25y+3f2n53fn5x5y2+f3n53fn5y3+.(6)k1,k2,k3yn+1=yn+h(1k1+2k2+3k3),yn+1h4yn+1h4=162hp3h3D3fn+3hqh12sp2h2D2fn5fn5y+12q2h22sphDfn52fn5x5y+2fnsphDfn52fn5y2+16q3h353fn5x3+3fn53fn5x25y+3f2n53fn5x5y2+f3n53fn5y3=162p3D3fn+312sp2qD2fn5fn5y+pq2sDfn52fn5x5y+fn52fn5y2+16q353fn5x3+3fn53fn5x25y+3f2n53fn5x5y2+f3n53fn5y3h4=162p3D3fn+123sp2qD2fn5fn5y+3pq2sDfn52fn5x5y+fn52fn5y2+163q3D3fnh4=h44!4(2p3+3q3)D3fn+123p2qs5fn5yD2f+243pq2s52fn5x5y+fn52fn5y2Dfn.(ii)y(xn+1)h4.yn=fn,yn=Dfn,yÊn=D2fn+5fn5yDfn,y(4)n=53fn5x+fn53fn5x25y+252fn5x5y5fn5x+fn5fn5y+2fn53fn5x25y+fn53fn5x5y2+2fn52fn5y25fn5x+fn5fn5y+f2n53fn5x5y2+fn53fn5y3+52fn5x5y+fn52fn5y25fn5x+fn5fn5y+5fn5y52fn5x2+fn52fn5x5y+5fn5x+fn5fn5y5fn5yfn52fn5x5y+fn52fn5y2=D3fn+5fn5yD2fn+352fn5x5y+fn52fn5y2Dfn+5f5y2Dfn.y(xn+1)h4h44!y(4)n=h44!D3fn+5fn5yD2fn+352fn5x5y+fn52fn5y2Dfn+5f5y2Dfn.(iii)y(xn+1)-yn+1.y(xn+1)-yn+1h44!D3fn+5fn5yD2fn+352fn5x5y+fn52fn5y2Dfn+5f5y2Dfn95:Runge2Kutta-h44!4(2p3+3q3)D3fn+123p2qs5fn5yD2fn+243pq2s52fn5x5y+fn52fn5y2Dfn=h44!(1-4(2p3+3q3))D3fn+(1-123p2qs)5fn5yD2fn+(3-243pq2s)52fn5x5y+fn52fn5y2Dfn+5f5y2Dfn.(7)(d)2p3=13p-3pq2,3q3=13q-2p2q.(c)2p3+3q3=13(p+q)-pq(2p+3q)=13(p+q)-12pq.(e)3pqs=1/6,(7)y(xn+1)-yn+1h44!1-413(p+q)-12pqD3fn+(1-2p)5fn5yD2fn(3-4q)52fn5x5y+fn52fn5y2Dfn+5f5y2Dfn.(8)(3).,[2],(1):(a),(b),(c),(d),(e)min|E|=|y(xn+1)-yn+1|,r+s=1,1+2+3=1,2p+3q=12,2p2+3q2=13,3pqs=16,0p1,0q1.(9).(8)y(xn+1)-yn+1h44!1-413(p+q)-12pqD3fn+(1-2p)5fn5yD2fn+(3-4q)52fn5x5y+fn52fn5y2Dfn+5f5y2Dfn.5f5y2Dfn=0,y(xn+1)xnTaylorh4y(xn+1)h4=h44!D3fn+5fn5yD2fn+352fn5x5y+fn52fn5y2Dfn+5fn5y2Dfn.(10)5fn5y=0,062005y(xn+1)-yn+1h44!1-413(p+q)-12pqD3fn+(3-4q)52fn5x5y+fn52fn5y2Dfn,y(xn+1)h4=h44!D3fn+352fn5x5y+fn52fn5y2Dfn.,1-43(p+q)+2pq(3-4q)=13,p(2-3q)=0,q=23;y(xn+1)-yn+119h44!y(4)n.p=14,(9)1=14,2=0,3=34,r=-13,s=43,yn+1=yn+14h(k1+3k3),k1=f(xn,yn),k2=fxn+14h,yn+14hk1,k3=fxn+23h,yn+29h(-k1+4k2).(11),Runge2Kutta,.:[1],,.[M].:,1995.[2],,.Runge2Kutta[J].,2001,26(1):1.OptimalAlgorithmofStronglyStableRunge2KuttainThree2stepOrderWUYi,ZHANGXiang2hui,XIONGChao2xi(DepartmentofBasicCourses,QinhuangdaoInstituteofTechnology,Qinhuangdao066004,China)Abstract:Thenumericalmethodsinfirstorderdifferenceequationandtheconvergenceandstabilityofnumericalsolutioninfirstorderdifferentialequationarestudiedinthispa2per.AccordintoRunge2kuttamethod,theoptimalcoefficientsaredeterminedbyoptimaltechnology,atthesametime,theoptimalalgorithmofstronglystableRunge2kuttainthree2steporderaregot.Keywords:Stabilityofnumerical;Stronglystable;Theoptimalcoefficients;Runge2Kutteinthree2steporder16:Runge2Kutta