Hybrid Methods for Solving Differential Equations

Transcription

1 Hybrid Methods for Solving Differential Equations Nicola Oprea, Petru Maior University of Tg.Mureş, Romania Abstract This paper presents and discusses a framework for introducing hybrid methods share with Runge-Kutta methods the property of utilizing data at points other than the step points. Thus we may regard the introduction of hybrid formulae as an important step into the no-man s land. A major advantage of hybrid methods is that they can possess remarkably small error constants. This methods are a class whose computational potentialities have probably not yet been fully exploited. Key words: Initial value problem, hybrid formula, difference operator, local truncation, predictor-corrector methods, linear multistep methods, weak stability 1 Introduction Consider the initial value problem for a single first-order differential equation: y = f(x, y), y(a) = y 0. (1.1) We seek a solution in the range a x b, where a and b are finite, and we assume that f satisfies the conditions which guarantee that the problem has a unique continuously differentiable solution, which we shall indicate by y(x). Consider the sequence of points {x n }, defined by x n = a + nh, n = 0, 1, 2,..., b a. The parameter h will always be h regarded as constant. An essential property of the majority of computational methods for the solution of (1.1) is that of discretization; that is: we seek an approximate solution, not on the continuous interval a x b, but on the discrete point set {x n }. Let y n be an approximation to the theoretical solution at x n, that is y(x n ) and let be f n = f(x n, y n ). We define a k-step hybrid formula to be a formula of the type: α j y = h β j f + h β ν f n+ν (1.2) where α k = 1, α j and β j are constants, α 0 and β 0 are not both zero, ν {0, 1,..., k}, and consistently with our previous usage, f n+ν = f(x n+ν, y n+ν ). In order to implement such a formula, even when it is explicit (has β k = 0) a special predictor to estimate y n+ν is necessary. Thus a hybrid formula, unlike a linear multistep method, can not be regarded as a method in its own right. We say that formula (1.1) is zero-stable if no root of the polynomial p(ρ) = α j ρ j

2 has modulus greater than one and if every root with modulus one is simple. We say that (1.1) has order r and error constant c r+1 if c 0 = c 1 = = c r = 0, c r+1 0, for a suitably differentiable test function y(x). We associate the difference operator L defined by: L[y(x); h] = α j y(x + jh) hβ j y (x + jh) hβ ν y (x + νh) (1.3) Expanding the test function y(x + jh) and its derivative y (x + jh) as Taylor series about x and collecting terms in (1.3) gives: L[y(x); h] = c 0 y(x) + c 1 hy (x) + + c q h q y (q) (x) +..., (1.4) where c q are constants. The principal local truncation error at x n+k is then defined to be c r+1 h r+1 y (r+1) (x n ). Order and error constant are most conveniently established by defining the constants c q as follows: c 0 = D 0 and we obtain: c 1 = D 1 + td 0 c 2 = D 2 + td 1 + t2 2! D 0. c q = D q + td q tq q! D 0. D 0 = α 0 + α α k D 1 = tα 0 + (1 t)α 1 + (2 t)α (k t)α k (β 0 + β β k + β ν ),. (1.5) D q = 1 q! [( t)q α 0 + (1 t) q α 1 + (2 t) q α (k t) q 1 α k ] (q 1)! [( t)q 1 β 0 + +(1 t) q 1 β 1 + (2 t) q 1 β (k t) q 1 β k + (ν t) q 1 β ν ], q = 2, 3,... The order of (1.2) is r if and only if, for any t, D 0 = D 1 = = D r = 0, D r+1 0; c r+1 is then equal to D r+1. Let us consider the case of the implicit two-step formula of class (1.2), it is: y n+2 + α 1 y n+1 + α 0 y n = h(β 2 f n+2 + β 1 f n+1 + β 0 f n + β ν f n+ν ), ν {0, 1, 2} (1.6) We start by choosing: α 1 = (1 + a), α 0 = a, 1 a < 1. (1.7) This choice ensures that the method has order at least zero and is zero-stable. From (1.5) with t = 1, we obtain:

3 D 0 = 0 D 1 = a + 1 (β 0 + β 1 + β 2 + β ν ), D 2 = 1 2! (a + 1) [ β 0 + β 2 + (ν 1) β ν ], D 3 = 1 3! ( a + 1) 1 2! [β 0 + β 2 + (ν 1) 2 β ν ], D 4 = 1 4! (a + 1) 1 3! [ β 0 + β 2 + (ν 1) 3 β ν ], (1.8) D 5 = 1 5! ( a + 1) 1 4! [β 0 + β 2 + (ν 1) 4 β ν ], D 6 = 1 6! (a + 1) 1 5! [ β 0 + β 2 + (ν 1) 5 β ν ] Excluding a, which we intend to retain for the moment, there are five undetermined parameters: β 0, β 1, β 2, β ν and ν. We might thus expect to attain order five by solving the set of five equations: D 1 = D 2 = D 3 = D 4 = D 5 = 0. However these equations, besides being non-linear in ν, are subject to the constraint ν {0, 1, 2}, and we can not easily say whether a solution exists. Gragg and Stetter have proved the following result: With the hybrid formula (1.2) associate the polynomials p(ρ) = k α j ρ j and σ(ρ) = β j ρ j, where k k, p satisfies the condition of zero-stability and p(1) = 0. The only cases of interest are k = k corresponding to (1.2) being implicit and k = k 1 corresponding to it being explicit. Then for all (but a finite number) of inadmissible such polynomials p, there exist uniquely a polynomial σ, a constant β ν and a real number ν {0, 1, 2,..., k } such that the order of (1.2) is at least k + 3. A corresponding result for conventional linear multistep methods guarantees, for any p subject to p(1) = 0, the unique existence of a σ of degree k such that the order of the associated linear multistep method is at least k +1. Thus Gragg and Stetter s result shows that, with certain exceptions, we can utilize both of the new parameters ν and β ν, we have introduced to raise the order of hybrid formula (1.2) to two above that attainable by a linear multistep method having the same left-hand side and the same value for k. Note that this is done without affecting control over zero-stability. Returning to our implicit two-step formula (1.6), it is convenient to set ν 1 = s, where s { 1, 0, 1}, in (1.8).

4 Solving D 1 = D 2 = D 3 = D 4 = D 5 = 0, we obtain: β 2 = 1 12 (5 + a) a s 1 β 1 = 2 3 (1 a) a s β 0 = 1 12 (5a + 1) a s + 1 β ν = a 4 s(s 2 1) s = a 1 + a (1.9) There are certain special cases which must be excluded from this solution. If a = 1, then s is infinite. From the last of the equations (1.9) it follows that the zero-stability condition 1 a < 1, assumed in (1.7) is satisfied if and only if s > 0. However, we still have to satisfy the requirement s { 1, 0, 1} and hence the case s = 1, or equivalently a = 7/23, must be excluded. So we must add to (1.9) the conditions: 1 < a < 1, a 7/23. (1.10) The method (1.6) with coefficients given by (1.7), (1.9) and (1.10) has order five and is zero-stable this is a result consistent with Gragg and Stetter s general theorem. From (1.8) and (1.9) the error constant is seen to be: D 6 = 1 2 6! (1 + a)(3s2 1) (1.11) where s is given in terms of a by the last of (1.9). Now we ask whether there remains an allowable choice for a which causes the order of (1.6) to rise to six. From (1.11), this will be the case if a = 1, s = 1/ 3 or s = 1/ 3. The first of these possibilities is excluded by (1.10) and the last is excluded by the condition of zero-stability which, as we have seen, is equivalent to s > 0. The choice s = 1/ 3 is however allowable, and gives the following values for the parameters in (1.6), which now has order six: α 2 = 1 β 2 = ( )(3 3) α 1 = β 1 = 8 3 3( ) α 0 = β 0 = 3 1 ( )(3 + 3) (1.12) ν = β ν = Error constant is 8 3 D 7 = ( , ) 9 7!

5 The spurious root of p for this formula is , 040 corroborating that the formula 3 is indeed zero-stable. We have thus succeeded in deriving a zero-stable implicit two-step hybrid formula of order six. With only seven parameters we could not hope for order greater then six, we have in fact achieved zero-stability with no sacrifice in order. We can obtain one of Butcher s formulae from equations (1.6), (1.7) and (1.9) by seting s = 1, it is: 2 y n (32y n+1 y n ) = h 93 (15f n f n+1 f n + 64f n+ 3 ) 2 a zero-stable formula of order five, whose error constant is D 6 = Hybrid predictor-corrector methods The value of f n+ν which must be computed before any of the hybrid formulae may be obtained by using of a special predictor of the form: y n+ν + α j y = h β j f (2.13) followed by the evaluation f n+ν = f(x n+ν, y n+ν ). The order of the predictor (2.13) is defined to be r, if for a suitable differentiable test function y(x), y(x + νh) + [α j y(x + jh) hβ j y (x + jh)] L[y(x); h] = = c 0 y(x) + c 1 hy (x) + + c q h q y (q) (x) +..., where c 0 = c 1 =... c r = 0, c r+1 0; the error constant is then c r+1 and the principal local truncation error c r+1 h r+1 y (r+1) (x n ). It is convenient to define order through the constants D q defined as follows: D 0 = α 0 + α α + 1, D 1 = tα 0 + (1 t)α (k 1 t)α + ν t (β 0 + β β ) D q = 1 q! [( t)q α 0 + (1 t) q α (k 1 t) q α + (ν t) q ] 1 (q 1)! [( t)q 1 β 0 + (1 t) q 1 β (k 1 t) q 1 β ], q = 2, 3,... (2.14) The order of (2.13) is then r if and only if, for any t, D 0 = D 1 = = D r = 0, D r+1 0; D r+1 then equals c r+1. We note that (2.13) is of quite a different form from (1.2), thus we cannot talk of the zero-stability of (2.13), since the expression is no longer a polynomial. α j ρ j + ρ ν

6 If the hybrid formula (1.2) is explicit (β k = 0), then we may clearly form an algorithm on the basis of (1.2) and (2.13) alone. However, if (1.2) is implicit, then we need addition to (2.13) a conventional predictor to calculate a value y [0] n+k. We shall indicate by P H an application of an explicit hybrid formula of class (1.2), by C H an application of an implicit formula of the same class and by P ν an application of a formula of class (2.13). P k indicates an application of a conventional explicit linear multistep method and E an evaluation of f in terms of known arguments. Thus, in any algorithm, P H or C H must be preceded by P ν E. The use of (2.13) together with an explicit formula of class (1.2) may be described as a P ν EP H E mode; if is used an implicit formula of class (1.2), then are possible the modes: P ν EP k EC H E and P ν EP k EC H. These modes are formally defined below: P ν EP H E : P ν : y n+ν + α j y = h β j f E : f n+ν = f(x n+ν, y n+ν ) P H : α j y = h β j f + hβ ν f n+ν P ν EP k EC H E : P ν : y n+ν + E : f n+k = f(x n+k, y n+k ); α j y [1] E : f n+ν = f(x n+ν, y n+ν ) P k : y [0] n+k + αjy [1] ( E : f [0] n+k = f x n+k, y [0] C H : n+k ) β j f [1] β j f [1] α j y [1] = hβ kf [0] n+k + h E : f [1] n+k = f(x n+k, y [1] n+k ); P ν EP k EC H : P ν : y n+ν + α j y [1] E : f n+ν = f(x n+ν, y n+ν ) P k : y [0] n+k + αjy [1] ( ) E : f [0] n+k = f x n+k, y [0] n+k C H : α j y [1] β j f [1] + hβ νf n+ν β j f [0] β j f [0] β j f [0] + hβ νf n+ν (2.15) (2.16) (2.17) As in the case of conventional predictor-corrector algorithms, we frequently have to use predictors whose stepnumbers exceed that of the hybrid formula P H or C H. We take account of this by removing the constraint that α 0 and β 0 do not both vanish and replace it by the restriction that not all of: α 0, β 0, α 0, β 0, α0 and β0 vanish.

7 All of the algorithms discussed by Gragg and Stetter fall into one or other of these three modes. Butcher introduces a new possibility which turns out to have certain advantages. For example: the mode defined by (2.16), after the stage P ν E has been concluded, it would be possible to replace the conventional predictor P k by one which involved, on its right-hand side, the value f n+ν, which is available. This means that replacing P k by an explicit hybrid formula of class (1.2) that is P H. Thus we obtain the mode P ν EP H EC H E which is given by (2.16) with the P k stage replaced by P H, where P H : y [0] n+k + αjy [1] We can similarly construct the mode P ν EP H EC H. β j f [1] + hβ νf n+ν. (2.18) 3 Weak stability of hybrid methods It is easy to find stability polynomials for the various hybrid predictor-corrector modes we have described. For the mode P ν EP H E defined by (2.15), if we introduce the notation: p(ρ) = α j ρ j ; σ(ρ) = β j ρ j, p(ρ) = (3.19) α j ρ j ; σ(ρ) = β j ρ j the stability polynomial is found to be: Π(r, h) = p(r) hσ(r) + hβ ν [p(r) hσ(r)] Stability polynomials for the other modes we have discussed can all be deduced from those for the most complicated modes considered, namely the P µ EP ν P H EC H (E) modes, which may be formally defined as follows: P µ : y n+µ + P ν : y n+ν + P H : y [0] n+k + C H : α j y [1] α j y [1] αjy [1] β j f [t] β j f [t] + hβ µf n+µ β j f [t] + hβ µf n+µ α j y [1] = hβ kf [0] n+k + h β j f [t] + hβ νf n+ν (3.20) where t = 1 when the mode includes the final evaluation (E) and t = 0 when it does not. We define the following polynomials:

8 p(ρ) = α j ρ j, σ(ρ) = β j ρ j p(ρ) = α j ρ j, σ(ρ) = β j ρ j p (ρ) = p(ρ) = αjρ j, σ (ρ) = βj ρ j α j ρ j, σ(ρ) = β j ρ j (3.21) The polynoms p and σ are associated with P H in (3.19), but with C H in (3.21). For the mode P µ EP ν EP H EC H E the stability polynomial is: Π(r; h) = p hσ + hβ k (p hσ ) + hβ ν (p hσ) + h 2 (β k β µ + β ν β µ )( p h σ), while, for the P µ EP ν EP H EC H mode, it is: Π(r; h) = r k (p hσ) + h(p σ pσ ) + hβ ν [r k (p hσ) + h(p σ pσ )] + + h 2 β ν β µ [r k ( p h σ) + h(p σ pσ )] + h 2 β µ[( pσ p σ) + hβ ν (p σ pσ)], where we have written p for p(r), σ for σ(r) etc. On comparing (3.20) with (2.16), (2.17) and (2.18), it is evident that we recover the modes P ν EP H EC H (E) from (3.20) by setting: β µ = β ν, β µ = 0, p(ρ) p(ρ), σ(ρ) σ(ρ) since the P µ and P ν steps are then identical. If we further set we recover the modes P ν EP k EC H (E). β ν = 0 4 Comparison with linear multistep and Runge-Kutta methods If a comparison on the basis of attainable order for a given number of function evaluations were to be made, hybrid methods would not fare particularly well, since most of the hybrid modes we have discussed call for three or four evaluations per step. However, such an approach ignores a major advantage of hybrid methods namely that they can possess remarkably small error constants. Meaningful comparisons must therefore rest on numerical experiments. To date there have been no really comprehensive numerical tests comparing hybrid with other methods. Hybrid methods present considerable difficulties when the steplength has to be changed and their implementation has not yet been developed to a stage comparable with that of conventional predictor-corrector methods.

9 References [1] L.C. Piccinini, G. Stampacchia, G. Vidossich: Ordinary Differential Equations in R n. Problems and Methods, Springer-Verlag (New-York, Berlin, Heidelberg, Tokyo). [2] J.D. Lambert: Computational Methods in Ordinary Differential Equations, John Wiley & Sons, (London, New York, Sydney, Toronto). [3] Corneliu Berbente, Sorin Mitran, Silviu Zancu: Metode numerice, Editura tehnică, Bucureşti.