A CLASS OF CONTINUOUS HYBRID LINEAR MULTISTEP METHODS FOR STIFF IVPs IN ODEs

ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I. CUZA DIN IAŞI (S.N.) MATEMATICĂ, Tomul LVIII, 0, f. A CLASS OF CONTINUOUS HYBRID LINEAR MULTISTEP METHODS FOR STIFF IVPs IN ODEs BY R.I. OKUONGHAE Abstract. In this paper, we present a class of continuous hybrid linear multistep methods (CHLMM) for stiff initial value problems (IVPs) in ordinary differential equations (ODEs). The constuction of these methods are based on the approach of collocation and interpolation. The interval of absolute stability of the method is investigated, using the root locus method. Numerical results of the methods solving stiff IVPs in ODEs are compared with that from the state-of-the-art Ode5s Matlab ODEs code. Mathematics Subject Classification 000: 65L05, 65L06. Key words: continuous hybrid linear multistep methods, collocation and interpolation, initial value problem, stiff stability, root locus.. Introduction Consider the numerical solution of stiff IVPs () y = f(x, y), y(x 0 ) = y 0, a x b by a class of continuous hybrid LMM (CHLMM) () (3) k y(x n +(t + )h)= α j (t)y n+j +α v (t)y n+v +hβ v (t)f(x n+v, y n+v ), t [, k ], 0 v k k y(x n + vh) = αj(t )y n+j + hβk (t )f n+k, v = t +, t = k 3,

40 R.I. OKUONGHAE where y n+j is the numerical approximation to the exact solution y(x n+j ), f n+j = f(x n+j, y n+j ), {α j (t), j = 0()k }, α v (t), {αj (t ), j = 0()k}, β v (t), and βk (t), are continuous coefficients in t presumed to be real and satisfying the normalization condition α k (t) =, αv(t ) =, x = x n+ +th, t [a, b] and h = x n+ x n is a fixed mesh size. The problem of stiffness in most ordinary differential equations (ODEs) has posed a lot of computational difficulties in many practical application modeled by ODEs. Stiffness affects the efficiency of numerical methods. Here, we present a class of continuous Hybrid linear multistep methods for stiff IVPs in ODEs. Hybrid LMM was first proposed by []. Other authors are, [,, 3, 4, 6, 9, 0,, 3, 4, 6, 9, 0,, ]. In fact, the numerical solution of () by () through collocation and interpolation methods have been well studied in the literature, see for example [3], [5], [5], [7, 8], [6], [6], [4], and [7, 8]. The interval of absolute stability of the CHLMM is investigated using the root locus method discussed in [0, ] and [3], whose application can be found in [6] and [4] instead of the equivalent boundary locus plot in [7] and [9]. The local truncation error for () and (3) are nicely given as (4) (5) k L.T.E = [y(x n + (t + )h) α j (t)y(x n + jh) α v (t)y( n +vh) hβ v (t)y (x n + vh)] and k L.T.E = [y(x n + vh) αj(t )y(x n + jh) hβk (t )y (x n + kh)]. The order for () and the expression for y n+v in the function f n+v in () are p = k+ and p = k+ respectively. Effective implementation of () demand the use of the Newton iterative scheme y [s+] n+k s = 0,,,..., where, F (y [s] = y[s] n+k F (y [s] n+k ) F (y [s] n+k ), n+k ) is the Jacobian matrix of the vector systems of the method. In particular, for k step the nonlinear equation F (y [s] n+k ) = 0. The parameter v is incorporated to provide off step collocation point x n+v in an open interval (x n+k, x n+k ) and v = k, where k is the step number of the scheme. Formula () is zero stable for fixed step size, h case for k 7. For k 8, no stable process appear to exist. See the root locus plots of the methods in section 4. The motivation to derive the hybrid method () is the fact that, it offers the means to by pass the Dahlquist

3 A CLASS OF CONTINUOUS HYBRID LINEAR MULTISTEP METHODS 4 order barrier for A-stable conventional LMM and the fact that continuous solution of the IVPs in ODEs can be obtained. The proposed continuous hybrid LMM in () consists of the addition of terms α v (t)y n+v to the left hand side of the One-leg hybrid LMM in [9] k α jy n+j = hβ v f(x n+v, y n+v ). Implementation of () required us to compute first y n+v in (3) so that the terms y n+v and f n+v in () could be evaluated. Considerations as to how this might be done appear in section 5 in this paper. The Outline of this paper is as follows. We start with the construction of the continuous hybrid LMM in () of k + in section. Section 3 deals with the derivations of the continuous hybrid predictor y n+v in the function f n+v of the method in (3). In section 4, we determined the stiff stability of the methods, using the root locus. Finally, in section 5, result of numerical experiments on some stiff test systems are presented and compared with Ode5s code from MATLAB ODE suite in [5].. Derivation of the continuous hybrid linear multistep methods The solution of the IVPs in () is assumed to be the polynomial k+ (6) y(x) = a j x j where {a j } k+ (8) we have are the real parameter constants to be determined. From k+ (7) y (x) = f(x, y) = ja j x j Collocating (7) at x = x n+v and interpolating (9) at x = x n+j, j = 0()k and x = x n+v, we obtain the linear system of equations (8) j= x n x n... x k+ n x n+ x n+... x k+ n+....... x n+k x n+k... x k+ n+k x n+v x n+v... x k+ n+v 0 x n+v... (k + )x k n+v a 0 a. a k a k a k+ = y n y n+. y n+k y n+v f n+v.

4 R.I. OKUONGHAE 4 Solving equation (8) for a js and substituting the resulting values into (6) with t = (x x n+ )/h and setting x = x t+ on the left hand side of (6) yield the values of the continuous coefficients {α j } k (t), α v(t), β v (t) respectively. Fixing t = k into the continuous coefficients {α j } k (t), α v (t), β v (t) for a fixed k, give the values of the discrete coefficients of the method in () for k 7. For example, see Table in appendix A for the continuous coefficients for method () for k 7. Table. in appendix A shows explicitly the discrete coefficients for method () for k 7. 3. The derivation of the continuous hybrid predictor Similarly, the corresponding hybrid predictor (9) y(x n + vh) = k αj(t)y n+j + hβk (t)f n+k, v = t +, t = k 3/ for y(x n+v ), and f(x n+v ) in () are obtained from the polynomial interpolant k+ (0) y(x n+v ) = b j x j. where {b} k+ are the real parameter constants to be determined. Following the same procedure in section, the unknown continuous coefficients of the hybrid predictors in () are obtained. After some simplifications, we obtained a class of continuous hybrid predictors from (). Table 3 in appendix B below shows the continuous coefficients of the predictor (3) for k 7. Table 4. in appendix B gives the discrete coefficients for the predictor (3) for k 7. 4. Stability of the methods by plotting the root locus In this section, we investigate the stability properties of the family of the continuous hybrid linear multistep method (CHLMM) in () using the root locus plot discussed in [0, ]. On substituting the hybrid solution (3) y n+v at point x n+v into the continuous hybrid LMM for a fixed k and t, and applying the resultant method on the scalar test problem y = λy, Re(λ) < 0, we obtain the continuous hybrid LMM stability polynomials to

5 A CLASS OF CONTINUOUS HYBRID LINEAR MULTISTEP METHODS 43 be k k k π(r, z) = r k α j r j α v ( αjr j + zβk rk ) zβ v ( αjr j + zβk rk ), () z = λh. Plotting r j (z) against z reveals the interval of absolute stability for the methods. The general form of the stability plot is given below in figure. Method () is said to be stable respectively, if 0 r j (z) where, r j (z), j = 0()k are roots of the polynomial in () with root r j (z) = been simple. Plotting the root locus of π(r, z) = 0, it is observed that the methods in () are stiffly stable for k 7. The graphs in figures -9 below show the loci and thus the interval of absolute/stiff stability of each method for a fixed value of k 7. The case of k 8 are stiffly unstable, see figure 9 and Table 4.3 respectively. r j Region of Absolute Instability Region of Absolute Instability - Z Region of Absolute Instability 0 Region of Absolute Instability Z Region of Absolute Instability Region of Absolute stability Figure : The root locus form of the region of absolute stability/stiff stability. See [0]

44 R.I. OKUONGHAE 6 Root Locus Plots for the Continuous Hybrid Multistep Methods in () r j r j. 0. 5. 0 0. 5 0 5 0 5 0 5 0 Figure : Root Locus Plot for k 0. 0 0 5 0 5 0 5 0 Figure 3: Root Locus Plot for k r j r j. 0. 5. 0 0. 5 0 5 0 5 0 5 0 Figure 4: Root Locus Plot for k 3 r j 0. 0 0 5 0 5 0 5 0 Figure 5: Root Locus Plot for k 4 r j. 0. 5. 0 0. 5 0 5 0 5 0 5 0 Figure 6: Root Locus Plot for k 5 r j 0. 0 0 5 0 5 0 5 0 5 Figure 7: Root Locus Plot for k 6 r j.5.0 0.5 0 5 0 5 0 5 0 5 Figure 8: Root Locus Plot for k 7 0.0 40 30 0 0 0 0 0 Figure 9*: Root Locus Plot for k 8

7 A CLASS OF CONTINUOUS HYBRID LINEAR MULTISTEP METHODS 45 Table 4.: The continuous and discrete error constants of method(). k t Continuous Error Constant (Cp+(t)) Cp+ 0 4 ( + t)( + t) h 3 y (3) 4 96 t( 3t + 4t3 )h 4 y (4) 3 480 (3 t) t( + t )h 5 y (5) 4 3 880 (5 t) t( + t)( + t)t( + t)h 6 y (6) 0 5 4 060 (7 t) ( 3 + t)( + t)( + t)t( + t)h 7 y (7) 6 5 680 (9 t) ( 4 + t)( 3 + t)( + t)( + t)t( + t)h 8 y (8) 7 6 4550 ( t) ( 5 + t)( 4 + t)( 3 + t)( + t)( + t)t( + t)h 9 y (9) 8 7 45500 (3 t) ( 6 + t)( 5 + t)( 4 + t)( 3 + t)( + t)( + t)t( + t)h 0 y (0) 48 80 68 4 88 360 Table 4.: The continuous and discrete error constants of predictor(3). k t Continuous Error Constant (Cp+(t )) C p+. 3 3 5 4 7 5 9 6 7 3 8 6 t ( + t)h 3 y (3) 48 4 ( + t) t( + t)h 4 y (4) 8 0 ( + t) t( + t )h 5 y (5) 70 ( 3 + t) ( + t)( + t)t( + t)h 6 y (6) 7 307 5040 ( 4 + t) ( 3 + t)( + t)( + t)t( + t)h 7 y (7) 3 4030 ( 5 + t) ( 4 + t)( 3 + t)( + t)( + t)t( + t)h 8 y (8) 33 3768 36880 ( 6 + t) ( 5 + t)( 4 + t)( 3 + t)( + t)( + t)t( + t)h 9 y (9) 43 96608 368800 ( 7 + t) ( 6 + t)( 5 + t)( 4 + t)( 3 + t)( + t)( + t)t( + t)h 0 y (0) 43 56 048 644 where, Cp+ and C p+ in Table 4. and Table 4. implies Discrete Error Constants for the CHLMM () and Hybrid Predictor (3) respectively.

46 R.I. OKUONGHAE 8 Table 4.3: The step-number, scaled variable t and interval of absolute stability of z for the method in () and (3). k t Interval of Absolute Stability of z Order() Order(3) for CHLMM ()+ Predictor(3) 0 (, 0) (4, ) (, 0) (6, ) 3 3 3 (, 0) (7.46, ) 4 4 4 3 (, 0) (8.667, ) 5 5 5 4 (, 0) (9.7, ) 6 6 6 5 (, 0) (0., ) 7 7 7 6 (, 0) (.46, ) 8 8 8 7 Unstable 9 9 Table 4.4: The step-number, the range of t for which the CHLMM () and predictor (3) is zero-stable. k The range of t for which the CHLMM ()+ Predictor(3) is zero stable {t : tε[, ]} {t : tε(,.85) (,.6) (.78, )} 3 {t : tε(,.53) (0,.55) (.6, )} 4 {t : tε(,.9) (, 3.43) (3.545, )} 5 {t : tε(,.5075) (0, ) (.9,.46) (, 4.3754) (4.504, )} 6 {t : tε(, 0) (0., ) (.6, )} 7 {t : tε(0,.0) (, 3) (3.35, 3.83) (4, 6.96) (6.4569, )} 8 Unstable

9 A CLASS OF CONTINUOUS HYBRID LINEAR MULTISTEP METHODS 47 5. Numerical experiments In this section the implementation of the CHLMM in () discussed in sections and 3 of this paper on the stiff initial value problems will be considered. Problem : Linear problem in [7] 0. 0 0 0 y = 0 0 0 0 0 0 00 0 y, y(0) =, y(x) = 0 0 0 000 Problem : Nonlinear chemical problem in [7] and [5] y = 400y + 0 4 y y 3 3 0 7 y, y 3 = 3 0 7 y, y(0) = 0 0 e 0.x e 0x e 00x e 000x y = 0.04y + 0 4 y y 3, with x being the range [0,0] for problem [] and h = 0.000 for problem [], x 0(0.000)3. In solving the initial value problems above, set up the continuous form of the methods from the continuous coefficients table of interest, CHLMM in () for k = is () y(x n +(t+)h) = (+4t+4t )y n +( 4t 4t )y n+ +h(+3t+t )f n+ from table () in Appendix A. The local truncation error and the order for (4) is (3) C 3 (t) = ( + t)( + t) h 3 y (3) (x), p =. 4 Setting t = 0 in (4) gives the equivalent discrete form of the CHLMM in () to be (4) y n+ = y n + hf n+, p =

48 R.I. OKUONGHAE 0 Similarly, from table (3) in Appendix B, we obtained the equivalent discrete form of the continuous hybrid predictor in (3) for k = and t = respectively to be (5) y n+ = 4 y n + 3 4 y n+ h 4 f n+, p = It has been noted by [7], [9], [, 3], and [0], that linear multistep methods suitable for stiff ODEs must be implicit and must therefore require a scheme to resolve the implicitness of the methods. Applying discrete methods () and (3) respectively to the initial value problem above leads to solving implicit set of equations which demands the use of Newton Raphson iterative scheme, (6) y [s+] n+k = y[s] n+k F (y [s] n+k ) F (y [s] n+k ), s = 0,,,... where F (y [s] n+k ) is the Jacobian matrix of the vector systems of the method. In particular, for k = the nonlinear equation F (y [s] n+k ) = 0, where (7) F (y [s] n+ ) = y[s] n+ y n hf ( x n+, y [s] ) = 0 n+ (8) y [s] n+ = 4 y n + 3 4 y(x[s] n+ ) h 4 f(x n+, y [s] n+ ). In this regard y [0] n+ is given from the trapezoidal rule (9) y [0] n+ = y n + h (f n+ + f n ), s = 0,,,... as an initial guess for y n+ in (9). Let L be the Lispschitz constant of f(x, y) with respect to y. For non-stiff problems, where L is small, the step size is usually determined by accuracy conditions. However, for stiff problems where L is large, the step size is severely restricted by stability constraint. Terminations of the iteration (6) occur whenever we observe that y [s+] n+ y[s] n+ TOL, where TOL is the order of the unit round off error of the computer, which may be assumed by the user. However figure

A CLASS OF CONTINUOUS HYBRID LINEAR MULTISTEP METHODS 49 Numerical Solution of Prob []. Numerical Solution of Prob [] 0.8 y (x) 0.6 0.4 0. 0 3 4 5 6 7 8 9 0 x 0. 0 3 4 5 6 7 8 9 0 x Numerical Solution of Prob []. Numerical Solution of Prob [] CHLMM Ode5s 0.8 y 4 (x) 0.6 0.4 3 4 5 6 7 8 9 0 x 0. 0 0. 0 3 4 5 6 7 8 9 0 x [h] Figure 0: The plot of numerical solutions of Problem and Ode5s in [5]. 4 x 0 5 Numerical Solutions CHLMM Ode5s 3.5 3.5 y (x).5 0.5 0 0 3 x Figure : The plot of numerical solutions of y (x) of Problem. 0 and figure below show the plot of the numerical results of the methods when applied to the linear problem and nonlinear problem. Finally, in this paper, we have derived a class of continuous hybrid linear multistep methods () which is of order p = k+, and stiffly stable for k 7 using collocation and interpolation process. The root locus plot in figure 9 revealed that the instability of the methods in () set in when k 8. The numerical solution graphs in figure 0 and figure, of the methods in (4) and (5) coincide and show that the methods in () is compared with the state-of the-art of MATLAB ode5s code in [5], on Problem and Problem respectively.

50 R.I. OKUONGHAE Appendix A

3 A CLASS OF CONTINUOUS HYBRID LINEAR MULTISTEP METHODS 5

5 R.I. OKUONGHAE 4

5 A CLASS OF CONTINUOUS HYBRID LINEAR MULTISTEP METHODS 53

54 R.I. OKUONGHAE 6 Appendix B

7 A CLASS OF CONTINUOUS HYBRID LINEAR MULTISTEP METHODS 55

56 R.I. OKUONGHAE 8

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