Numerical Methods for Ordinary Dierential Equations Episode 14 pdf

GENERAL LINEAR METHODS 439 Definition 551A A general linear method (A, U, B, V ) is ‘inherently Runge– Kutta stable’ if V is of the form (551a) and the two matrices BA −XB and BU − XV + VX are zero except for their first rows, where X is some matrix. The significance of this definition is expressed in the following. Theorem 551B Let (A, U, B, V ) denote an inherently RK stable general linear method. Then the stability matrix M(z)=V + zB(I −zA) −1 U has only a single non-zero eigenvalue. Proof. Calculate the matrix (I − zX)M(z)(I −zX) −1 , which has the same eigenvalues as M(z). We use the notation ≡ to denote equality of two matrices, except for the first rows. Because BA ≡ XB and BU ≡ XV −VX, it follows that (I − zX)B ≡ B(I −zA), (I − zX)V ≡ V (I −zX) −zBU, so that (I − zX)M (z) ≡ V (I −zX). Hence (I − zX)M(z)(I − zX) −1 is identical to V , except for the first row. Thus the eigenvalues of this matrix are its (1, 1) element together with the p zero eigenvalues of ˙ V .  Since we are adopting, as standard r = p + 1 and a stage order q = p,itis possible to insist that the vector-valued function of z, representing the input approximations, comprises a full basis for polynomials of degree p.Thus,we will introduce the function Z given by Z =         1 z z 2 . . . z p         , (551b) 440 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS which represents the input vector y [n−1] =         y(x n−1 ) hy  (x n−1 ) h 2 y  (x n−1 ) . . . h p y (p) (x n−1 )         . (551c) This is identical, except for a simple rescaling by factorials, to the Nordsieck vector representation of input and output approximations, and it will be convenient to adopt this as standard. Assuming that this standard choice is adopted, the order conditions are exp(cz)=zA exp(cz)+UZ + O(z p+1 ), (551d) exp(z)Z = zB exp(cz)+VZ+ O(z p+1 ). (551e) This result, and generalizations of it, make it possible to derive stiff methods of quite high orders. Furthermore, Wright (2003) has shown how it is possible to derive explicit methods suitable for non-stiff problems which satisfy the same requirements. Following some more details of the derivation of these methods, some example methods will be given. 552 Conditions for zero spectral radius We will need to choose the parameters of IRKS methods so that the p × p matrix ˙ V has zero spectral radius. In Butcher (2001) it was convenient to force ˙ V to be strictly lower triangular, whereas in the formulation in Wright (2002) it was more appropriate to require ˙ V to be strictly upper triangular. To get away from these arbitrary choices, and at the same time to allow a wider range of possible methods, neither of these assumptions will be made and we explore more general options. To make the discussion non-specific to the application to IRKS methods, we assume we are dealing with n × n matrices related by a linear equation of the form y = axb − c, (552a) and the aim will be to find lower triangular x such that y is strictly upper triangular. The constant matrices a, b and c will be assumed to be non-singular and LU factorizable. In this discussion only, define functions λ, µ and δ so that for a given matrix a, λ(a) is unit lower triangular such that λ(a) −1 a is upper triangular, µ(a) is the upper triangular matrix such that a = λ(a)µ(a), δ(a) is the lower triangular part of a. GENERAL LINEAR METHODS 441 Using these functions we can find the solution of (552a), when this solution exists.Wehaveinturn δ(axb)=δ(c), δ  µ(a −1 ) −1 λ(a −1 ) −1 xλ(b)µ(b)  = δ(c), δ  λ(a −1 ) −1 xλ(b)  = δ  µ(a −1 )δ(c)µ(b) −1  , implying that x = δ  λ(a −1 )δ  µ(a −1 )δ(c)µ(b) −1  λ(b) −1  . (552b) Thus, (552b) is the required solution of (552a). This result can be generalized by including linear constraints in the formulation. Let d and e denote vectors in R n and consider the problem δ(axb − c)=0,xd= e. Assume that d is scaled so that its first component is 1. The matrices a, b and c are now, respectively n ×(n −1), (n −1) ×n and (n −1) ×(n −1). Partition these, and the vectors d and e,as a =  a 1 a 2  ,b=  b 1 b 2  ,d=  1 d 2  ,e=  e 1 e 2  , where a 1 is a single column and b 1 asinglerow. The solution to this problem is x =  e 1 0 e 2 − xd 2 x  , where x satisfies δ(ax  b − c) = 0, and a = a 2 ,  b = b 2 − d 2 b 1 , c = c − aeb 1 . Finally we consider the addition of a second constraint so that the problem becomes δ(axb − c)=0,xd= e, f x = g , where c is (n − 2) × (n −2) and the dimensions of the various other matrix and vector partitions, including the specific values d 1 = f 3 = 1, are indicated in parentheses a =  a 1 (1) a 2 (n−2) a 3 (1) (n−2)  b =    b 1 (n−2) (1) b 2 (n−2) b 3 (1)    d =    1 (1) (1) d 2 (n−2) d 3 (1)    e =    e 1 (1) (1) e 2 (n−2) e 3 (1)    f =  f 1 (1) f 2 (n−2) 1 (1) (1)  g =  g 1 (1) g 2 (n−2) g 3 (1) (1)  442 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS For both linear constraints to be satisfied it is necessary that f e = f Bd = g d. Assuming this consistency condition is satisfied, denote the common value of f e and g d by θ. The solution can now be written in the form x =    e 1 00 e 2 − xd 2 x 0 e 3 + g 1 − θ + f 2 xd 2 g 2 − f 2 xg 3    , where δ(ax  b − c)=0, with a = a 2 − a 3 f 2 ,  b = b 2 − d 2 b 1 , c = c − aeb 1 − a 3 g b + θa 3 b 1 . 553 Derivation of methods with IRK stability For the purpose of this discussion, we will always assume that the input approximations are represented by Z given by (551b), so that these approximations as input to step n are equal, to within O(h p+1 ), to the quantities given by (551c). Theorem 553A If a general linear method with p = q = r − 1=s − 1 has the property of IRK stability then the matrix X in Definition 551A is a (p +1)× (p +1) doubly companion matrix. Proof. Substitute (551d) into (551e) and compare (551d) with zX multiplied on the left. We find exp(z)Z = z 2 BAexp(cz)+zBUZ + VZ+ O(z p+1 ), (553a) z exp(z)XZ = z 2 XB exp(cz)+zXV Z + O(z p+1 ). (553b) Because BA ≡ XB and BU ≡ XV −VX, the difference of (553a) and (553b) implies that zXZ ≡ Z + O(z p+1 ). Because zJZ ≡ Z + O(z p+1 ), it now follows that (X − J)Z ≡ O(z p ), which implies that X − J is zero except for the first row and last column.  We will assume without loss of generality that β p+1 =0. GENERAL LINEAR METHODS 443 By choosing the first row of X so that σ(X)=σ(A), we can assume that the relation BA = XB applies also to the first row. We can now rewrite the defining equations in Definition 551A as BA = XB, (553c) BU = XV − VX+ e 1 ξ , (553d) where ξ =[ ξ 1 ξ 2 ··· ξ p+1 ] is a specific vector. We will also write ξ(z)=ξ 1 z + ξ 2 z 2 + ···+ ξ p+1 z p+1 . The transformed stability function in Theorem 551B can be recalculated as (I − zX)M(z)(I −zX) −1 = V + ze 1 ξ (I − zX) −1 , with (1, 1) element equal to 1+zξ(I − zX) −1 e 1 = det(I + z(e 1 ξ −X)) det(I − zX) = (α(z)+ξ(z))β(z) α(z)β(z) + O(z p+2 ), (553e) where the formula for the numerator follows by observing that X − e 1 ξ is a doubly companion matrix, in which the α elements in the first row are replaced by the coefficients of α(z)+ξ(z). The (1, 1) element of the transformed stability matrix will be referred to as the ‘stability function’ and denoted by R(z). It has the same role for IRKS methods as the stability function of a Runge–Kutta method. For implicit methods, the stability function will be R(z)=N(z)/(1 − λz) p+1 ,whereN(z) is a polynomial of degree p +1 givenby N(z)=exp(z)(1 − λz) p+1 −  0 z p+1 + O(z p+2 ). The number  0 is the ‘error constant’ and is a design parameter for a particular method. It would normally be chosen so that the coefficient of z p+1 in N (z) is zero. This would mean that if λ is chosen for A-stability, then this choice of  0 would give L-stability. For non-stiff methods, λ =0andN(z)=exp(z) −  0 z p+1 + O(z p+2 ). In this case,  0 would be chosen to balance requirements of accuracy against an acceptable stability region. In either case, we see from (553e) that N(z)=α(z)(β(z)+ξ(z))+ O(z p+1 ), so that ξ(z), and hence the coefficients ξ 1 , ξ 2 , , ξ p+1 can be found. Let C denote the (p +1)× (p +1) matrix with (i, j)elementequalto c j−1 i /(j − 1)! and E the (p +1)× (p +1) matrixwith (i, j)elementequalto 1/(j −i)! (with the usual convention that this element vanishes if i>j). We can now write (551d) and (551e) as U = C − ACK, V = E − BCK. 444 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS Substitute into (553d) and make use of (553c) and we find BC(I − KX)=XE −EX + e 1 ξ . (553f) Both I −KX and XE −EX + e 1 ξ vanish, except for their last columns, and (553f) simplifies to BC         β p β p−1 . . . β 1 1         =          1 1! 1 2! ··· 1 p! 1 (p+1)! −  0 0 1 1! ··· 1 (p−1)! 1 (p)! . . . . . . . . . . . . 00··· 1 1! 1 2! 00··· 0 1 1!                  β p β p−1 . . . β 1 1         . Imposing conditions on the spectrum of V implies constraints on B.This principle is used to derive methods with a specific choice of the vector β and the abscissa vector c. Rather than work in terms of B directly, we introduce the matrix  B = Ψ −1 B. Because  BA =(J + λI)  B, and because both A and J +λI are lower triangular,  B is also lower triangular. In the derivation of a method,  B will be found first and the method coefficient matrices found in terms of this as A =  B −1 (J + λI)  B, U = C −ACK, B =Ψ  B, V = E − BCK. To construct an IRKS method we need to carry out the following steps: 1. Choose the value of λ and  0 taking into account requirements of stability and accuracy. 2. Choose c 1 , c 2 , , c p+1 . These would usually be distributed more or less uniformly in [0, 1]. 3. Choose β 1 , β 2 , , β p . This choice is to some extent arbitrary but can determine the magnitude of some of the elements in the coefficient matrices of the method. 4. Choose a non-singular p × p matrix P used to determine in what way ˙ V has zero spectral radius. If δ is defined as in Subsection 552, then we will impose the condition δ(P −1 ˙ VP) = 0. It would be normal to choose P as the product of a permutation matrix and a lower triangular matrix. GENERAL LINEAR METHODS 445 5. Solve the linear equations for the non-zero elements of  B from a combination of the equations δ(P −1 ˙ Ψ  BC ˙ KP)=δ(P −1 ˙ EP)and  BC         β p β p−1 . . . β 1 1         =Ψ −1          1 1! 1 2! ··· 1 p! 1 (p+1)! −  0 0 1 1! ··· 1 (p−1)! 1 (p)! . . . . . . . . . . . . 00··· 1 1! 1 2! 00··· 0 1 1!                  β p β p−1 . . . β 1 1         . 554 Methods with property F There is a practical advantage for methods in which e 1 B = e p+1 A, e 2 B = e p+1 . A consequence of these assumptions is that β p =0. For this subclass of IRKS methods, in addition to the existence of reliable approximations hF i = hy  (x n−1 + hc i )+O(h p+2 ),i=1, 2, ,p+1, (554a) where y(x) is the trajectory such that y(x n−1 )=y [n−1] 1 ,thevalueofy [n−1] 2 provides an additional approximation hF 0 = hy  (x n−1 )+O(h p+2 ), which can be used together with the p + 1 scaled derivative approximations given by (554a). This information makes it possible to estimate the values of h p+1 y (p+1) (x n )andh p+2 y (p+2) (x n ), which are used for local error estimation purposes both for the method currently in use as well as for a possible method of one higher order. Thus we can find methods which provide rational criteria for stepsize selection as well as for order selection. Using terminology established in Butcher (2006), we will refer to methods with this special property as possessing property F. They are an extension of FSAL Runge–Kutta methods. The derivation of methods based on the ideas in Subsections 553 and 554 is joint work with William Wright and is presented in Wright (2002) and Butcher and Wright (2003, 2003a). 446 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 555 Some non-stiff methods The following method, for which c =[ 1 3 , 2 3 , 1] , has order 2:   AU BV   =              000 1 1 3 1 18 3 10 001 11 30 11 90 1 5 5 12 0 1 23 60 7 45 5 3 − 29 12 4 3 1 5 12 2 9 −24−1 00 0 3 −96 00 0              . (555a) This method was constructed by choosing β 1 = − 1 6 , β 2 = 2 9 ,  0 =0and requiring ˙ V to be strictly upper triangular. It could be interpreted as having an enhanced order of 3, but of course the stage order is only 2. The next method, with c =[ 1 4 , 1 2 , 3 4 , 1] , has order 3:                    0000 1 1 4 1 32 1 384 224 403 0001 − 45 806 − 45 3224 67 19344 1851 2170 93 280 001 − 3777 8680 − 681 6944 297 138880 305 364 5 28 5 12 0 1 − 473 1092 − 81 728 17 17472 305 364 5 28 5 12 0 1 − 473 1092 − 81 728 17 17472 000100 0 0 − 156 7 188 7 −20 8 0 52 7 1 7 − 1 28 − 512 7 584 7 − 160 3 16 0 568 21 4 7 − 1 7                    . (555b) For this method, possessing property F, β 1 = 1 2 , β 2 = 1 16 ,  0 =0.The3×3 matrix ˙ V is chosen so that δ(P −1 ˙ VP)=0,where P =    001 100 410    . GENERAL LINEAR METHODS 447 556 Some stiff methods The first example, with λ = 1 4 and c =[ 1 4 , 1 2 , 3 4 , 1] , has order 3:  AU BV  =                    1 4 00010 − 1 32 − 1 192 11 2124 1 4 001 130 531 − 11 8496 − 719 67968 117761 23364 − 189 44 1 4 0 1 − 130 531 183437 186912 283675 747648 312449 23364 − 4525 396 1 36 1 4 1 − 650 531 121459 46728 130127 124608 − 58405 7788 4297 132 − 475 12 15 1 125 236 510 649 − 733 20768 − 64 33 746 33 − 95 3 12 00 85 44 677 1056 − 8 3 4 3 4 3 0 00 0 13 24 −32 112 −128 48 00 0 0                    . (556a) This method was constructed with β 1 = − 1 4 , β 2 = β 3 = 1 4 ,  0 = 1 256 and δ( ˙ V )=0.Thechoiceof 0 was determined by requiring the stability function to be R(z)= 1 − 1 8 z 2 − 1 48 z 3 (1 − 1 4 z) 4 , which makes the method L-stable. The second example has order 4 and an abscissa vector [ 1 3 4 1 4 1 2 1 ]: A =         1 4 0000 − 513 54272 1 4 000 3706119 69088256 − 488 3819 1 4 00 32161061 197549232 − 111814 232959 134 183 1 4 0 − 135425 2948496 − 641 10431 73 183 1 2 1 4         , U =         1 3 4 1 4 1 24 0 1 27649 54272 5601 54272 513 108544 − 153 54272 1 15366379 207264768 756057 69088256 1620299 414529536 − 1615 3636224 1 − 32609017 197549232 929753 65849744 4008881 197549232 58327 27726208 1 − 367313 8845488 − 22727 2948496 40979 5896992 323 620736         , 448 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS B =         − 135425 2948496 − 641 10431 73 183 1 2 1 4 00001 2255 1159 − 47125 10431 447 61 − 11 2 7 2 25240 3477 − 192776 10431 6728 183 −20 8 9936 1159 − 239632 10431 3120 61 −24 8         , V =         1 − 367313 8845488 − 22727 2948496 40979 5896992 323 620736 00 0 0 0 0 − 28745 10431 − 1937 13908 117 18544 65 11712 0 − 141268 10431 − 2050 3477 − 187 2318 113 1464 0 − 216416 10431 − 452 3477 − 491 1159 161 732         . (556b) This property F method was constructed with β 1 = 3 4 , β 2 = 3 16 , β 3 = 1 64 ,  0 = 13 15360 and δ(P −1 ˙ VP)=0,where P =      0 001 1 000 8 100 16410      . The method is L-stable with R(z)= 1 − 1 4 z − 1 8 z 2 + 1 96 z 3 + 7 768 z 4 (1 − 1 4 z) 5 . 557 Scale and modify for stability With the aim of designing algorithms based on IRKS methods in a variable order, variable stepsize setting, we consider what happens when h changes from step to step. If we use a simple scaling system, as in classical Nordsieck implementations, we encounter two difficulties. The first of these is that methods which are stable when h is fixed can become unstable when h is allowed to vary. The second is that attempts to estimate local truncation errors, for both the current method and for a method under consideration for succeeding steps, can become unreliable. Consider, for example, the method (555b). If h is the stepsize in step n, which changes to rh in step n + 1, the output would be scaled from y [n] to (D(r)⊗I N )y [n] ,whereD(r)=diag(1,r,r 2 ,r 3 ). This means that the V matrix which determines stable behaviour for non-stiff problems, becomes effectively [...]... stability property of implicit Runge–Kutta methods BIT, 15, 358–361 Butcher J C (1977) On A-stable implicit Runge–Kutta methods BIT, 17, 375–378 Numerical Methods for Ordinary Differential Equations, Second Edition J C Butcher © 2008 John Wiley & Sons, Ltd ISBN: 978-0-470-72335-7 454 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS Butcher J C (1979) A transformed implicit Runge–Kutta method J Assoc... 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