A family of A-stable Runge Kutta collocation methods of higher order for initial-value problems

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1 IMA Journal of Numerical Analysis Advance Access published January 3, 2007 IMA Journal of Numerical Analysis Pageof20 doi:0.093/imanum/drl00 A family of A-stable Runge Kutta collocation methods of higher order for initial-value problems JESÚS VIGO-AGUIAR AND HIGINIO RAMOS Scientific Computing Group, Universidad de Salamanca, Salamanca, Spain [Received on 27 January 2006; revised on 7 December 2006] We consider the construction of a special family of Runge Kutta (RK) collocation methods based on intra-step nodal points of Chebyshev Gauss Lobatto type, with A-stability and stiffly accurate characteristics. This feature with its inherent implicitness makes them suitable for solving stiff initial-value problems. In fact, the two simplest cases consist in the well-known trapezoidal rule and the fourth-order Runge Kutta Lobatto IIIA method. We will present here the coefficients up to eighth order, but we provide the formulas to obtain methods of higher order. When the number of stages is odd, we have considered a new strategy for changing the step size based on the use of a pair of methods: the given RK method and a linear multistep one. Some numerical experiments are considered in order to check the behaviour of the methods when applied to a variety of initial-value problems. Keywords: Runge Kutta collocation methods; initial-value problems; A-stability.. Introduction Implicit Runge Kutta (RK) methods may be classified according to whether or not they are collocation methods (Lambert, 99). For ordinary differential equations (ODEs), collocation consists in choosing a polynomial and demanding that its derivative coincides on a set of collocation points with the vector field of the differential equation. In this paper, we present a new family of methods that fall into the class of RK collocation methods, where the internal stages are based on the Chebyshev Gauss Lobatto points, which are the zeros of certain Chebyshev polynomial of second kind. These Chebyshev polynomials were chosen because of their superior convergence rate properties and their special characteristics in relation to the approximation of functions (Fox & Parker, 968; Mason & Handscomb, 2003). They have been widely used in numerical analysis (Reddy & Weideman, 2005; Berrut & Trefethen, 200; Panowsky & Richardson, 988; Vigo-Aguiar & Ramos, 2003). In the solution of differential equations, the Chebyshev Gauss Lobatto points are often preferable to other sets of Chebyshev points because they include the end points t = ± (which after certain transformations will be related to the approximate values y n and y n+ of the true solution of the differential equation). We will explore some features of the new methods, specifically, the order and the favourable property of A-stability. This attribute, with the fact that they are implicit methods, is very promising in connection with their use for solving stiff initial-value problems. In Section 2, we develop the construction of the family of methods, and in Section 3, we present formulas for obtaining the coefficients. Section is devoted to studying some particular cases and the specific characteristics of the methods are summarized in Section 5. The connection between the new methods and the linear multistep methods is outlined in Section 6. In Section 7, we present a particular strategy for changing the step size when an odd number of stages jvigo@usal.es Corresponding author. higra@usal.es c The author Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

2 2of20 J. VIGO-AGUIAR AND H. RAMOS is taken, in addition to some details concerning the implementation. In Section 8, different problems already used in tests in the literature are considered in order to assess the stability characteristics of the methods and their use for solving stiff initial-value problems. 2. Derivation of the new family We are interested in the numerical solution of ODEs, possibly stiff, of the form { y (t) = f (t, y(t)), t [t 0, T ], y(t 0 ) = y 0, (2.) where f is assumed to satisfy the conditions to guarantee the existence of a unique solution of the initial-value problem. We consider a fixed step size, h, i.e. h = (T t 0 )/k, for some integer k > 0, and a set of equally spaced points on the integration interval given by t 0 < t < < t k = T, where we set t j = t 0 + jh, j =,...,k. Now, suppose we have an approximation of the solution of (2.) at the point t n, n < k, i.e. y n y(t n ). In order to obtain an approximation y n+ for y(t n+ ), we consider the general (s + )-stage RK method defined by the Butcher tableau c a a 2 a (s+) c 2 a 2 a 22 a 2(s+) c s+ a (s+) a (s+)2 a (s+)(s+) (2.2) b b 2 b s+ which may be written in the alternative form (Lambert, 99) s+ y n+ci = y n + h a ij f (t n + c j h, y n+c j ), i =,...,s +, (2.3) j= s+ y n+ = y n + h b i f (t n + c i h, y n+ci ), i= where, as usual, y n+ci stands for the approximation of the true values y(t n + c i h). Now, in order to determine values for the coefficients a ij in the method, we set b i = a (s+)i, i =,...,s +, which makes the method stiffly accurate (Hairer & Wanner, 996). For c j, we take c j+ = ξ j = 2 ( + α j), j = 0,,...,s, (2.) where the α j are the Chebyshev Gauss Lobatto points given by α j = cos(θ j ), θ j = (s j)π, j = 0,,...,s. (2.5) s

3 A-STABLE RUNGE KUTTA COLLOCATION METHODS 3of20 We note that these points may be obtained as a projection of equally spaced points over the unit circle onto the interval [, ]. Moreover, they are the extrema on [, ] of the Chebyshev polynomial of the first kind T s (t) (Fox & Parker, 968). Let z(t) be a sufficiently many times differentiable function and define the linear operators associated with the equations in (2.3) by s+ L i (z(t), h) = z(t + ξ i h) z(t) h a ij z (t + ξ j h) (2.6) for i =,...,s +. Proceeding in the same way as is done for linear multistep methods, we expand z(t + ξ i h) and z (t + ξ j h) about t and collect powers in h to obtain where j= L i (z(t), h) = C i z () (t)h + C i2 z (2) (t)h 2 +, (2.7) C ip = ξ p i p! s+ (p )! j= a ij ξ p i (2.8) for i =,...,s +, p N. Note that we have a total number of (s +)(s +) coefficients a ij to be determined. If we impose that the intermediate steps have order (s + ), i.e. the coefficients C ip for i =,...,s +, p =,...,s + vanish, it results in an algebraic system of (s + )(s + ) equations in the (s + )(s + ) unknowns a ij. That is, we have the same number of equations and unknowns, and solving this system we will obtain the required values for the RK method in (2.2). We note that the methods could have been obtained using the theory of block-gams methods described in Brugnano & Trigiante (998), or in Iavernaro & Mazzia (998), taking variable step sizes computed using the Chebyshev Gauss Lobatto points. In this way, a system of (s + )-step linear methods is obtained, and imposing that each formula has maximum order (s + ), the coefficients are uniquely determined. 3. Solving the algebraic system If s is large, solving the resulting linear algebraic system may be an impossible task, even with the help of a computer algebra system. However, we note that what we really have are (s +) uncoupled systems, each of them formed by (s + ) equations with the same number of unknowns. These systems have the form a j + a j2 + +a j (s+) = ξ j, a j ξ 0 + a j2 ξ + +a j (s+) ξ s = ξ 2 j 2, for j =,...,s +. a j ξ s 0 + a j2ξ s + +a j (s+)ξ s s = ξ s+ j s +,. (3.)

4 of20 J. VIGO-AGUIAR AND H. RAMOS THEOREM 3. For j =,...,s +, each system of equations given by Bv j = Ψ j, where v j = ξ 0 ξ ξ s B = ξ 2 0 ξ 2 ξs 2, ξ0 s ξ s ξs s a j a j2. a j (s+) ξ j ξ 2 j /2, Ψ j =,. ξ s+ j /(s + ) has a unique solution. Proof. The result follows immediately from elementary algebra because the matrix B is a Vandermonde matrix, and all the values ξ 0,...,ξ s being distinct B is nonsingular, and each system of linear equations in (3.) has a unique solution. A closed form of the solution for these systems is presented, which has been obtained by considering the inverse of the Vandermonde matrix. THEOREM 3.2 The solution of system (3.) may be expressed in the form a jk = s l + ω (k )lξ l+ j, j =,...,s +, k =,...,s +, l=0 where ω ij = ( ) s j sk=0,k i (ξ i ξ k ) σ s j(ξ 0,...,ξ i,ξ i+,...,ξ s ), i, j = 0,...,s, (3.2) and σ i (x,...,x m ) refers to the elementary symmetric polynomial of degree i in the variables x,...,x m. Proof. Since the matrix B is nonsingular, we have v j = B Ψ j, and using the explicit formula for the inverse of the Vandermonde matrix (Higham, 2003, p. 26), which is given by B = (ω ij ) i, s j=0 where the ω ij are defined as in (3.2), the result follows immediately. REMARK 3.3 The methods we have obtained could also be categorized as collocation RK methods, taking as collocation points the Chebyshev Gauss Lobatto points in (2.5). Imposing the conditions for obtaining the coefficients as in Lambert (99), we obtain a similar system of equations as in (3.).

5 A-STABLE RUNGE KUTTA COLLOCATION METHODS 5of20. Examples of the methods For conclusions concerning the order of the methods presented in the above sections, additional results are needed. We present these results as corollaries of two theorems that we reproduce from Hairer et al. (993, p. 22). THEOREM. An implicit RK method with all c i different and of order at least p is a collocation method iff C(p): p j= a ij c q j = cq i q, i =,...,p, q =,...,p, is true. COROLLARY.2 The RK method given by the Butcher tableau in (2.2) whose coefficients satisfy (2.) and (3.) is a collocation method. Proof. This follows immediately observing that by setting the values c i as in (2.), the equations in (3.) reduce to the condition C(s + ). THEOREM.3 Let M(t) = p i= (t c i) and suppose that M is orthogonal to polynomials of degree (r ), i.e. 0 M(t)t q dt = 0, q =,...,r; then the RK collocation method obtained from the c i has order (p + r). COROLLARY. When s is even, the RK collocation method described in the above sections has order (s + 2). Proof. We consider the function M(t) = s i=0 (t ξ i ) and q =, and calculate the integral in the above theorem using ξ i = ( + cos θ i )/2 and the transformation z = 2t. We have 0 M(t)dt = 0 s (t ξ i )dt = i=0 2 s+2 i=0 = 2 s+2 s (z cos θ i )dz (z 2 )q(z)dz, (.) where q(z) = s i= (z cos θ i). The roots of q(z) are the extrema of the Chebyshev polynomial of the first kind T s (z), which has leading coefficient 2 s (Fox & Parker, 968), and thus we may write q(z) = T s (z) s2 s. Finally, as T s (z) is an even function, T s (z) is an odd function, and so is (z 2 )q(z), and the integral in (.) is zero. According to the theorem, the order of the method is (s + ) + = s + 2.

6 6of20 J. VIGO-AGUIAR AND H. RAMOS. Case s = For the simplest case, after calculating the coefficients using Theorem 3.2, the RK method in (2.2) results in /2 /2 (.2) /2 /2 Arranging the method in a different form, this is the trapezoidal rule which is known to have second order and to be A-stable (Lambert, 99, p. 225)..2 Case s = 2 In this case, after calculating the coefficients using Theorem 3.2, the RK method is given by /25/2 /3 /2 /6 2/3 /6 (.3) /6 2/3 /6 which is the Runge Kutta Lobatto IIIA method, which has order four and is also A-stable (Butcher, 2003)..3 Case s = 3 Now, for s = 3, we obtain the Butcher tableau / 59/576 7/288 7/288 5/576 3/ 3/6 5/32 9/32 3/6 /8 /9 /9 /8 (.) /8 /9 /9 /8 For s 3, as far as we know, the methods in this article do not correspond to any method that has appeared in the literature. To obtain the order of the (s + )-stage method, we check the conditions for order (s + ) from the Butcher theory (Butcher, 2003), and it turns out that the method has at least order (s + ) (for this task, it is useful to consider the package implementing the Butcher theory in the computer algebra system Mathematica). On the other hand, if we apply the RK method in (2.2) to the test equation y (t) = λy(t), λ C, a one-step difference equation results of the form y n+ = R s (λh)y n, where the stability function R s (z), z = λh, is given by R s (z) = det[i za+ zebt ], det[i za]

7 A-STABLE RUNGE KUTTA COLLOCATION METHODS 7of20 where e is the (s + )-vector e = (,...,) T, I is the (s + )-identity matrix, A is the matrix A = (a ij ) s+ i, j= and b = (b,...,b s+ ) T. The region of the complex z-plane in which R(z) isthe region of absolute stability of the method. For s = 3, the stability function is R 3 (z) = z + 38z2 + 3z z + 38z 2 3z 3, and, for z in the left-half complex plane, we have R 3 (z), which means that the method is A-stable.. Case s = The following and subsequent cases follow a study similar as in the case above. For s =, the Butcher tableau that defines the method is / For the stability function, we have R (z) = z + 20z2 + 22z 3 + z z + 20z 2 22z 3 + z Once more, R (z) on the left-half complex plane, and thus the method is A-stable. With respect to the order, using the Mathematica system, we have corroborated the order conditions from the Butcher theory for the method to have order at least five. However, as has occurred for the case s = 2, it follows from Corollary. that the order is in fact six. 30 (.5).5 Case s = 5 The tableau that defines the method takes the form (9 5) 75 7( 7+2 5) (9+ 5) (9+ 5) (7+2 5) (9 5) (9 5) 75 2(9+ 5) 75 2(9+ 5) 75 2(9 5) 75 50

8 8of20 J. VIGO-AGUIAR AND H. RAMOS The order conditions for order six may be verified and not for order seven, thus the method has order six. In this case, the stability function is given by R 5 (z) = z z z z + 5z z z 2 296z z 5z 5, and R 5 (z) on the left-half complex plane, indicating that the method is A-stable..6 Case s = 6 Now, the matrix A is given by (8+7 3) , and the vectors c and b are c = [ 0, 2 3,, 2, 3, 2 + T 3, ], [ b = 70, 8 63, 8 35, 82 35, 8 35, 8 63, ]. 70 where It follows from Corollary. that the method has order eight. Now, the stability function is given by R 6 (z) = N 6(z) D 6 (z), N 6 (z) = z z z z + 6z 5 + 3z 6, D 6 (z) = z z z z 6z 5 + 3z 6, and on the left-half complex plane R 6 (z), so this method is also A-stable.

9 A-STABLE RUNGE KUTTA COLLOCATION METHODS 9of20 5. Characteristics of the methods The first characteristic is about the stage order because as Lambert (99) has indicated, the accuracy of the internal stages can be as important as the final point. From the very construction of the methods, it follows that the accuracy of the internal stages is at least O(h s+2 ), i.e. y n+ci = y(t n + c i h) + O(h s+2 ). Moreover, as the first row of the matrix A consists of zeros, the first stage coincides with the initial value. And due to the requirement of stiff accuracy, the last stage coincides with the expression for the final point, which implies that no further function evaluation is necessary to obtain y n+. As we have established above, the methods are A-stable. This feature is in accordance with the fact that the imaginary axis is the contour of the stability region for symmetric RK methods (Butcher, 987; Brugnano & Trigiante, 998), as the methods in this article are. This last feature is obtained directly because the numbers c,...,c s+ in (2.) are symmetrically distributed on the interval [0, ], i.e. c i = c s+2 i for i =,...,s +. This means that the numerical result obtained by applying the above methods with step size h to y n+ is exactly the initial value y n (Hairer et al., 993, p. 222). Moreover, we observe that for the above symmetric methods, the stability function has the form R s (z) = P s(z) P s ( z), where P s (z) is a polynomial of degree s that may be obtained by P s (z) = M (s+) () + M (s) ()z + + M ()z s, where M(z) = s i=0 (z ξ i ) with ξ i as in (2.) (Nørsett, 975). Finally, the above methods are not symplectic because they do not satisfy the simple algebraic relation (Sanz-Serna, 998) b i a ij + b j a ji b i b j = 0, i, j =,...,s Connection between RK and linear multistep methods Surprisingly, the RK methods in this article may also be obtained from linear multistep formulas, providing in this way a direct connection between RK methods and linear multistep methods. The relation with multistep formulas could have been obtained also from the block-gams formulation already mentioned, but in what follows, we give a complete information about the particular multistep formulas. As we have pointed out, the simple case for s = corresponds to the linear multistep method known of a trapezoidal rule. For s = 2, the method in (.3) consists of a system of two equations given by y n+ 2 = y n + h 2 (5 f n + 8 f n+ 2 f n+ ), y n+ = y n + h 6 ( f n + f n+ 2 + f n+ ). (6.)

10 0 of 20 J. VIGO-AGUIAR AND H. RAMOS Looking carefully at the first equation in (6.), it is the two-step Adams Moulton formula for the equation y = f (x, y) given by ( 5 y n+ = y n + h 2 f n f n ) 2 f n, (6.2) where we have to set x n = t n + h, x n = t n + h/2, x n+ = t n and h = h/2. Now, if we take the two-step Milne Simpson formula for the equation y = f (x, y) given by ( y n+ = y n + h 3 f n+ + 3 f n + ) 3 f n, (6.3) setting x n = t n, x n = t n + h/2, x n+ = t n + h and h = h/2, we recover the second of the formulas in (6.). For s 3, the situation is a little more complicated as the linear multistep methods to be considered must be formulated in their variable step-size form. We will sketch out the procedure in the case s = 3, but for larger values of s, the way to pass from the RK formulation to the multistep formulation is very similar. The method corresponding to the partitioned matrix in (.) may be expressed by means of the system of three equations: y n+ = y n + h 576 y n+ 3 = y n + 3h 6 y n+ = y n + h 8 (59 f n + 9 f n+ f n f n+ ), ( f n + 0 f n+ + 6 f n+ 3 f n+ ), (6.) ( f n + 8 f n+ + 8 f n+ 3 + f n+ ). The first of these equations may be obtained from the three-step Adams Bashforth formula with variable steps for the problem y = f (x, y), where the grid points have to be taken as x n 2 = t n + h, x n = t n + 3h/, x n = t n + h/, x n+ = t n, and the step sizes are h n = x n x n 2 = h/, h n = x n x n = h/2, h n+ = x n+ x n = h/. Using the above points and step sizes, the three-step Milne Simpson formula in its variable step-size version results in the second of the equations in (6.). And finally, for the same values, the generalized variable-step Milne Simpson formula of three steps (Henrici, 962) leads to the last equation in (6.).

11 A-STABLE RUNGE KUTTA COLLOCATION METHODS of Implementation details 7. Strategy for the step-size selection As is usual for the RK methods, we might consider the use of an embedded pair of methods for the step-size selection. For each method of the family, we would look for an RK method of the form c A that satisfies that the order conditions have order p s, where A and c are the same as in the chosen RK method of the family. For example, for the method with s =, we could take ˆb = (0, /3, /3, /3, 0) which results in a fourth-order method. Thus, we would have an approximation of y(t n + h) by an RK method of lower order given by s+ ˆb ŷ n+ = y n + h ˆb i f (t n + c i h, y n+ci ). (7.) i= The difference between the approximations could be taken as an estimation for the local truncation error, err, given by s+ err = y n+ ŷ n+ = h (b i ˆb i ) f (t n + c i h, y n+ci ), (7.2) i= which behaves like O(h p+ ) for h 0. But the use of the embedded pair of methods described above would be very costly in terms of the number of function evaluations considering the observation in Section 5 about the last stage. Instead of this, we propose a different approach to obtain an estimate for the local truncation error when s is even. In this situation, we take into account the fact that ξ s/2 = /2, and consider the linear multistep method of second order (which is the two-step backward differentiation formula with step size h/2; Hairer et al., 993) given by hf n+ = 3y n+ + y n y n+/2. (7.3) From this formula, if we assume that z(t) is the true solution of Problem (2.), we have hf(t n+, z(t n+ )) = 3z(t n+ ) + z(t n ) z(t n+/2 ) + O(h 3 ), (7.) where t n+ and t n+/2 are abbreviations, respectively, for t n + h and t n + 2 h. Now, if y n+/2 is the approximate value at the intermediate stage obtained with an RK method of the family described in Section 2, assuming the localization hypothesis (Lambert, 99), y n = z(t n ), y n+/2 = z(t n+/2 ), after expanding in Taylor series about (t n+, y n+ ) the function on the left-hand side of (7.), we may write [ h f (t n+, y n+ ) + δ f ] δy (t n+, y n+ )(z(t n+ ) y n+ ) = 3z(t n+ ) + y n y n+/2 + O(h 3 ), (7.5) where δ f/δy denotes the Jacobian matrix.

12 2 of 20 J. VIGO-AGUIAR AND H. RAMOS On subtracting 3y n+ from both sides of (7.5) and rearranging, we get ( 3I m h δ f ) δy (t n+, y n+ ) (z(t n+ ) y n+ ) = hf(t n+, y n+ ) y n +y n+/2 3y n+ +O(h 3 ), (7.6) where I m stands for the identity matrix of order m. Finally, the estimate for the local truncation error may be obtained from the formula z(t n+ ) y n+ = err + O(h 3 ), where and M is the matrix err = M (hf(t n+, y n+ ) y n + y n+/2 3y n+ ) (7.7) M = 3I m h δ f δy (t n+, y n+ ). Note that with this strategy, the extra cost for changing the step size consists just in one more function evaluation per step but as this value will be used in the next step, there is no extra cost after all. Our numerical experiments confirm that using this procedure we can obtain great accuracy. Once we have derived an estimate for the local error, the standard step-size prediction (Shampine & Gordon, 975; Henrici, 962) leads to ( ) atol /3 h new = τ h old (7.8) err for a given tolerance, atol, where τ is a safety factor. In order to avoid excessive computations, we also consider the common strategy that if the new step size satisfies k h old h new k 2 h old, with, say, k =.0 and k 2 =.5, then we retain h old for the following step. 7.2 Other implementation issues Since the first row in the matrix of the method consists of zeros, we can rewrite the method in (2.3) in the form Y i = ĀF(Y j ) + δ n, (7.9) where Ā is the s s matrix obtained from the Butcher tableau with both the first row and the first column suppressed, Y i = (y n+c2,...,y n+cs+ ) T, F(Y j ) = ( f (t n + c 2 h, y n+c2 ),..., f (t n + c s+ h, y n+cs+ )) T, (7.0) and δ n = (y n + ha 2 f (t n, y n ),...,y n + ha (s+) f (t n, y n )) T contains the known terms that involved the previous computed solution.

13 A-STABLE RUNGE KUTTA COLLOCATION METHODS 3 of 20 To solve the implicit system (7.9), there are different approaches (Amodio & Brugnano, 997; Cooper & Butcher, 983; Iavernaro & Mazzia, 998). In our implementation, we solve the system by means of Newton s method taking as initial guess the values (computed at no additional cost) given by yn+c 0 i = y n + hc i f (t n, y n ), i = 2,...,s +. (7.) Observe that y n+ = y n+cs+, and so no further function evaluation is necessary to obtain y n+. Once the solution y n+ has been computed, the step-size selection strategy in Section 7. determines the new value h new according to the formula in (7.8). 8. Numerical illustration To check the numerical behaviour when used to solve stiff and nonstiff systems, we have applied the above methods to a variety of well-known problems which have appeared in the literature. We have considered a variable step-size version of the above (s + )-stage RK method, that will be named VSCRK s, where the error estimation was performed using the technique described in Section 7.. First, we have taken a problem from the test set in Mazzia & Iavernaro (2003) used in tests elsewhere, and then a few more problems that have appeared in different articles concerning specific methods for initialvalue problems. 8. The Robertson problem This classical problem that models the kinetics of a chemical reaction (Hairer & Wanner, 996) consists of a system of three equations given by y (t) = 0.0y (t) + 0 y 2 (t)y 3 (t), y 2 (t) = 0.0y (t) 0 y 2 (t)y 3 (t) y 2 (t) 2, (8.) y 3 (t) = 3 07 y 2 (t) 2, with initial conditions y (0) = and y 2 (0) = y 3 (0) = 0. This special system, as is typical for problems arising in chemical kinetics, has a small, very quick initial transient. It has been integrated on the interval [0, 0 ]. The reference solution at the end of the integration interval has been taken from the test set in Mazzia & Iavernaro (2003): y (t f ) = , y 2 (t f ) = , y 3 (t f ) = For this problem, we have considered a step-size controller that uses a relative tolerance, rtol, and an absolute tolerance, atol (Hairer et al., 993, p. 67). We impose that the error on each step satisfies componentwise err i sc i, where sc i = atol + rtol max{ yn i, yi n+ }. As a measure of the error, we take err ˆ = max { vi sc i },

14 of 20 J. VIGO-AGUIAR AND H. RAMOS where the v i are the components of the vector that appeared in (7.7), v = M (hf(t n+, y n+ ) y n + y n+/2 3y n+ ). The new step size is obtained as ( h new = τ h old err ˆ ) /d, where d = n(log 0 (atol) + log 0 (rtol)) and τ is a safety factor as in (7.8) that for this problem has been established as τ =.25 in order to avoid excessive steps. The results obtained with the new method are presented in Table with some of the data for different codes that appear in Mazzia & Iavernaro (2003). The parameters listed in the table are the prescribed relative and absolute tolerances, rtol, atol, the initial step size, h 0, the total number of steps, nstep, the number of function evaluations, feval, a measure of the error given by the scd factor, scd = log 0 max i { yiexact (t f ) y icomputed (t f ) y iexact (t f ) where t f is the final point at the integration, and finally, the number of Jacobian evaluations when solving the system by means of full or simplified Newton methods, jac. The number of LU-decompositions in all the codes is similar to the number of Jacobian evaluations (except for DASSL and VODE, for details see Mazzia & Iavernaro, 2003). Using the Newton method, it is possible to leave the Jacobian frozen for a few steps. We have indicated this possibility with J = k which gives the number of iterations required before updating the Jacobian (J = refers to the full Newton scheme). In Fig., we show the efficiency curves for these codes, where we have plotted the polygonals joining the points (log 0 (feval), scd), i.e. the relative error (scd factor) versus the computational cost measured by the logarithm of the number of function evaluations. TABLE Results for Problem (8.) Method rtol atol h 0 nstep feval scd jac VSCRK (J = ) VSCRK (J = 3) VSCRK (J = 6) DASSL MEBDFI VODE RADAU },

15 A-STABLE RUNGE KUTTA COLLOCATION METHODS 5 of 20 FIG.. Efficiency comparison curves for Problem (8.). 8.2 A Prothero Robinson equation We consider the Prothero Robinson equation { y (t) = λ(y(t) g(t)) + g (t), t [0, 0], y(0) = 0, (8.2) with λ = 0 6, g(t) = sin(t), and the exact solution given by y(t) = sin(t). This inhomogeneous problem is a particular case of the family of scalar equations proposed in Prothero & Robinson (97) and constitutes a stiff problem (since the eigenvalue is λ = 0 6 ). In Table 2, we present the results obtained with the method for s = 6 in this article for different step sizes, h = /2 n, where in column Err appears the maximum of the absolute error over the integration interval given by Err = max j {log 0 y exact (t j ) y computed (t j ) }, (8.3) and feval refers to the number of evaluations of the function in the right-hand side of the differential equation. In Fig. 2, we present in a double logarithmic scale, the results obtained with the methods VSCRK 2 and VSCRK in this paper, and the best results in a recent article by Jackiewicz et al. (200) when using a two-step L-stable W-method, which we name WJ. TABLE 2 Results for Problem (8.2) n feval Err(h)

16 6 of 20 J. VIGO-AGUIAR AND H. RAMOS FIG. 2. Efficiency curves for Problem (8.2). 8.3 A mildly stiff linear system For the next test problem, we consider a well-known classical system (Baker, 989; Stabrowski, 997). It is a mildly stiff problem composed of two first-order equations { y (t) = 998y (t) + 998y 2 (t), y 2 (t) = 999y (8.) (t) 999y 2 (t), with initial values y (0) = and y 2 (0) =, and exact solution given by the sum of two decaying exponential components { y (t) = e t 3e 000t, y 2 (t) = 2e t + 3e 000t. The stiffness ratio is : 000, and the problem is solved numerically on the interval [0, 0]. The local relative error has been set to 0 6. In Table 3, we present the results of test runs with the BGH stiff solver in Hall & Watt (976) and the version of the Gear method in Stabrowski (997), both of them belonging to the class of backward differentiation formula methods. For the method in this article, we have chosen s =, 6, 8 and initial step h 0 = 0. The parameters considered are the number of function evaluations, feval, and the total number of integration steps, nstep. Note that in this case the resulting system of algebraic equations is linear and the Jacobian is constant, so we have used a full Newton scheme. The exact and discrete solutions are presented in Fig. 3 on the time domain [0, 0.0]. TABLE 3 Results for Problem (8.) Method feval nstep BGH stiff Gear type VSCRK 70 VSCRK VSCRK

17 A-STABLE RUNGE KUTTA COLLOCATION METHODS 7 of 20 FIG. 3. Exact and discrete solutions using VSCRK for Problem (8.). Up: component y (t). Down: component y 2 (t). 8. The Brusselator system This nonstiff problem (Hairer et al., 993) consists of the two equations { y (t) = + y 2 (t)y 2(t) y (t), y 2 (t) = 3y (t) y 2 (t)y 2(t), (8.5) with initial values y (0) =.5 and y 2 (0) = 3, and has been integrated on the interval [0, 20]. In Table, the results obtained with the VSCRK method are presented. The definitions of the parameters are those in Section 8., and now AbsErr refers to the absolute error at the final point measured in the norm. The initial step has been taken in all cases as h 0 = 0, and the strategy for estimating the local error is the same as that in Section A discretization problem by the method of lines The last problem (Butcher, 2003) is intended to show that our method is applicable for solving stiff problems arising from semi-discretization of a partial differential equation using the method of lines. We consider the diffusion equation u t = 2 u, (t, x) [0, ] [0, ], (8.6) x2

18 8 of 20 J. VIGO-AGUIAR AND H. RAMOS TABLE Results for Problem (8.5) M atol nstep feval jac AbsErr VSCRK (J = ) VSCRK (J = 3) VSCRK (J = ) VSCRK (J = 3) with initial value u(0, x) = a sin( 2x) sin(x) and boundary conditions given by u(t, 0) = 0 and u(t, ) = a exp( 2t) sin( 2) exp( t) sin(), where a = cos( 2)/ 2 cos(2 /2 ). The spatial discretization results in an approximating system of ODEs in time after replacing the partial derivatives with respect to x in (8.6) evaluated at x j = j/(n + ) by the second-order central differences 2 u x 2 (x j, t) y j+ 2y j + y j ( x) 2, where x = /(N + ) and ( ) j y j (t) = u t,, j =,...,N, N + and taking into account that from the initial conditions, it is y 0 (t) = u(t, 0) and y N+ (t) = u(t, ). The resulting system is y (t) = My(t) + v(t), (8.7) FIG.. Error versus step size for Problem (8.7).

19 A-STABLE RUNGE KUTTA COLLOCATION METHODS 9 of 20 where M = (N + ) and v(t) = (N + ) 2 (y 0 (t), 0,...,0, y N+ (t)), as in Butcher & Rattenbury (2005). We integrate this problem using a constant step size h = /n and taking exactly n steps (n = 0, 00, 500). In Fig., we have plotted the results in a double logarithmic scale, comparing the method described in this article when s = 2, with the almost RK method of fourth order in Butcher & Rattenbury (2005) and the two-stage Gauss method. This problem was also integrated in a variable step-size implementation which resulted in very high accuracy for a rather small number of steps. 9. Conclusions In this paper, the construction of an implicit family of methods, appropriate for solving stiff initial-value problems, is described. The resulting methods are A-stable and exhibit the favourable characteristics of being symmetric and stiffly accurate. It can be easily checked that in the context of general linear methods (Butcher, 996), they are inherently RK stable. We have obtained explicit formulas for the coefficients of the methods for all orders. When an odd number of stages is taken, a new strategy for selecting the step size has been developed. By numerical examples, we show that these methods are promising and indicate that they may be competitive with other methods that are frequently used for stiff differential equations. Acknowledgements The authors wish to thank the JCYL project SA 02/0 and the MICYT project BMF for financial support. They also want to express their gratitude to the anonymous referees for their careful reading and for improving the manuscript. REFERENCES AMODIO, P.&BRUGNANO, L. (997) A note on the efficient implementation of implicit methods for ODEs. J. Comput. Appl. Math., 87, 9. BAKER, L. (989) C Tools for Scientist and Engineers. New York: McGraw-Hill. BERRUT, J.P.&TREFETHEN, L. N. (200) Barycentric Lagrange interpolation. SIAM Rev., 6, BRUGNANO, L.&TRIGIANTE, D. (998) Solving Differential Problems by Multistep Initial and Boundary Value Methods. Amsterdam, The Netherlands: Gordon & Breach. BUTCHER, J. C. (987) The Numerical Analysis of Ordinary Differential Equations. New York: Wiley.

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