-Stable Second Derivative Block Multistep Formula for Stiff Initial Value Problems

Transcription

1 IAENG International Journal of Applied Mathematics, :3, IJAM 3_7 -Stable Second Derivative Bloc Multistep Formula for Stiff Initial Value Problems (Advance online publication: 3 August )

2 IAENG International Journal of Applied Mathematics, :3, IJAM 3_7 with a =, a3 = + 3, a = 7 + a = 3 + 7, a 6 = + 8, a = + [ C = a 3 = + 3, a = 7 y n, y n+ +, a = a 6 = + 8 B = [a, a, a, a 3, a, a ] T ] T, y n+, y n++, hf n+, h g n+ solving the system of equation () for variable B = [a, a, a, a 3, a, a ] T and substituting in (3), we obtain a continuous multistep formula y(x) = where α j (x)y n+j + j= α vj (x)y n+vj + hβ (x)f n+ j= = +h δ (x)g n+ () α (x) = 7 x + 96 x 7 x3 + 9 x 38 x α v (x) = ( 6 ( + 36 x ( + + ( 8 + x 3 + ( x x 8 ) x α (x) = x + 78 x 777 x3 + 7 x 3 α v = ( ( + ( 8 ( x x + x 3 + ( 8 x β = x + x 3 x3 + 6 x 3 x δ = x 3 7 x x 9 x + x ) x Evaluating () at x = x n+ yields the main method, while differentiating () and evaluating at x = {x n+, x n+, x n++ } together, yields the bloc method represented in bloc matrix finite difference form where A= ( AY m = BY m + hcf m + h DG m (6) ) ( 9 ( ( 3 8 ( ( ( 6 9 ( ( B = ( 3 ( ( 3 C = ( 8 ( ( D = ( 8 +. The -dimensional vector Y m, Y m, F m and G m have collocation points specified as, Y m = [y n+v, y n+, y n+v, y n+ ] T, Y m = [y n v, y n, y n v, y n ] T, F m = [f n+v, f n+, f n+v, f n+ ] T, G m = [g n+v, g n+, g n+v, g n+ ] T., (Advance online publication: 3 August )

3 IAENG International Journal of Applied Mathematics, :3, IJAM 3_7 3 Analysis of the Method The analysis of the L(α)-stable bloc multistep method is presented in this section. Numerical Properties such as Order and Error constant, consistency, stability and convergence are investigated. Order and Error Constant Let the individual linear multistep method with Chebyshev collocation points be associated with the formula L[y(x n ; h)] = j= α j y(x + jh) + α vj y(x + v j h) hβ j y (x + jh) hβ vj y (x + v j h) h δ y (x n + h) (7) where y(x) is an arbitrary smooth function on [a, b]. Expanding (8) with Taylor series expansions of y(x + jh), y(x + v j h), y (x + jh), y (x + v j h) and y (x + h), j =, v,, v,,..., v, to obtain the expression L[y(x n ; h)] = C y(x)+c hy (x)+ +C p h p y (p) (x)+ where C i are vectors in the form, Using the appropriate coefficients in (6), methods are of order p = [,,, ] T with error constants C 6 = [.38 3, 9.,.88, 6.39 ]. T Consistency Since the bloc multistep method is of order p =, therefore it is consistent. Henrici [6] Zero Stability of Bloc Multistep Method Applying the bloc multistep method (6) to the test problem y = λy, with z = λh, solving the characteristic equation det ξ (A Cz Dz ) B = for ξ at z, the roots {,,, } of the resulting equation are less than or equal to, therefore the numerical method is zero-stable. Convergence C = C = C = C q = α j + j= jα j + j= (! α vj (8) j= v j α vj β j + j= j= j= β vj (9) j= j α j + ) j= v j α v ( j j= jβ j + ) j= v () jβ vj δ ( q! (q )! j= jq α j + ) j= vq j α v ( j j= jq β j + ) j= vq j β vj (q )! q δ q =,,,, p. () Since the Bloc multistep method is consistent and zero-stable, we can safely assert the convergence of the new method. (Henrici [6]) Region of Absolute Stability of new method Solving characteristic equation det ξ (A Cz Dz ) B = for ξ, we obtain the stability function as + 7z + z + z 3 R(z) = + 68z z + z 3 z + z () Definition 3. The bloc multistep method with Chebyshev collocation points (6) and the associated linear difference operator is said to be of order p if, C = C = C = = C p =, C p+. () Definition 3. The term C p+ is called the Error Constant (EC) and the local truncation error for the method is given by, t n+ = C p+ h p+ y (p+) x n + O(h (p+) ). (3) Solving (), we obtain the stability region S = [(, ) (., )]. The bloc multistep method is A -Stable and satisfies A(α)-Stability with stiff stability properties α = 89.8, D =.66 and y =.. Hence, The method is Stiffly Stable. The region of absolute stability is presented in Figure. Test for L(α)-Stability The numerical method is A(α)-stable and lim z R(z) =, we say that bloc multistep method is L(α)-Stable. (Advance online publication: 3 August )

4 IAENG International Journal of Applied Mathematics, :3, IJAM 3_7 Table : Problem.: Maximum Absolute Error: max<i<n y (x) y, n for h = n n New Method GBDF8 ATBM7 (Rate) (Rate) (Rate) (.99) (6.).8 7 (7.8). 3 (.99) (7.).8 9 (7.6) (.) 3. 9 (.) (6.6) 9. (6.8) (8.36).69 (7.9) Figure : RAS of SDBDF for = Experimental Problems Problem.: A linear stiff problem The linear system of 3 first order ordinary differential equations solved by Ainfenwa [], Brugnano and Trigiante [3] and Ramos and Garcia-Rubio [] given by, y = y + 9y y3, y = 9y y + y3, y3 = y y + y3, y () =, y () =, y () =, () on the interval < x < is solved with the newly derived bloc multistep method. We compare the maximum absolute errors ( y(x) yn ) on the interval < x < with the Adams Type Bloc method of Ainfenwa [] of order p = 7 (ATBM7) and Generalized Bacward Differentiation formula of Brugnano and Trigiante [3] (GBDF8), n =,,, 3 and for nuusing step lengths h = n merical solution of y(x). The order of the methods are also verified by calculating the rate of convergence with the formula errh Rateh = log, errh where errh is the maximum absolute error at step length h. Also in the range x, AbsErr(tf ) in [] is obtained by the new method in comparison with the CBDF of degree s = in Ramos and Garcia-Rubio [] and the following results are presented Remar.: Clearly from Table, it can be seen that the new method even though it is of order p =, performs better that the ATBM7 and the GBDF8, both of orders 7 and 8 respectively. Also, the rate of convergence of the new method conforms almost exactly with the order of our methods unlie the ATBM7 and GBDF8. Table shows that the new method is comparable with the CBDF in []. Numerical results also show that Table : Problem.: Numerical Results in comparison with CBDF in the range x Steps New Method Rate CBDF Rate the new method is consistent with order of the method as the step size decreases. Problem.: Cash []. We also consider the integration of the stiff system using the problem whose Jacobian matrix J has imaginary eigenvalues given by y = αy βy + (α + β )e t, y = βy αy + (α β )e t, y () =, y () =, t, (6) It is noted that for any given value of parameter α and β, J is the matrix, α β, β α with eigenvalues of J as α±iβ and the required solution is y (t) y (t) = e t, = e t, For the case α = and β = with a fixed step size (Advance online publication: 3 August )

5 IAENG International Journal of Applied Mathematics, :3, IJAM 3_7 Acnowledgement (Advance online publication: 3 August )

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