Numerical Methods for Ordinary Dierential Equations Episode 14 pdf

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 + V X are zero except for their fir

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 + V X 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

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

Since we are adopting, as standard r = p + 1 and a stage order q = p, it is 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

which represents the input vector

Assuming that this standard choice is adopted, the order conditions are

exp(cz) = zA exp(cz) + U Z + O(z p+1 ), (551d)

exp(z)Z = zB exp(cz) + V Z + 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 thesame requirements Following some more details of the derivation of thesemethods, 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 widerrange 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

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.

Using these functions we can find the solution of (552a), when this solutionexists We have in turn

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 Rn 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

, d =

1

,

where a1 is a single column and b1a single row

The solution to this problem is

$

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

a = a2− a3f2, b = b2− d2b1, c = c − aeb1− a3g b + θa3b1.

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 thequantities 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 = z2BA exp(cz) + zBU Z + V Z + O(z p+1 ), (553a)

z exp(z)XZ = z2XB exp(cz) + zXV Z + O(z p+1 ). (553b)

Because BA ≡ XB and BU ≡ XV − V X, the difference of (553a) and (553b)

We will assume without loss of generality that β = 0

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

where ξ = [ ξ1 ξ2 · · · ξ p+1] is a specific vector We will also write

ξ(z) = ξ1z + ξ2z2+· · · + ξ p+1 z p+1 The transformed stability function inTheorem 551B can be recalculated as

(I − zX)M(z)(I − zX) −1 = V + ze

1ξ (I − zX) −1 ,

with (1, 1) element equal to

1 + zξ(I − zX) −1 e1=det(I + z(e1ξ− X))

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 , where N (z)

is a polynomial of degree p + 1 given by

N (z) = exp(z)(1 − λz) p+1 0zp+1 + O(z p+2 ).

0is 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

0 would give L-stability

For non-stiff methods, λ = 0 and N (z) = exp(z) 0zp+1 + O(z p+2) In

0 would be chosen to balance requirements of accuracy against anacceptable 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 ξ1, ξ2, , ξ p+1 can be found

Let C denote the (p + 1) × (p + 1) matrix with (i, j) element equal to

Substitute into (553d) and make use of (553c) and we find

1 2! · · · 1

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:

0 taking into account requirements of stabilityand accuracy

2 Choose c1, c2, , c p+1 These would usually be distributed more or less

5 Solve the linear equations for the non-zero elements of B from a

combination of the equations δ(P −1Ψ ˙BC ˙ KP ) = δ(P −1 EP )and˙

1 2! · · · 1

554 Methods with property F

There is a practical advantage for methods in which

e1B = e p+1 A,

e2B = 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 reliableapproximations

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 ) = y1[n −1] , the value of y [n −1]

2provides an additional approximation

as for order selection

Using terminology established in Butcher (2006), we will refer to methodswith this special property as possessing property F They are an extension ofFSAL Runge–Kutta methods

The derivation of methods based on the ideas in Subsections 553 and 554 isjoint work with William Wright and is presented in Wright (2002) and Butcherand Wright (2003, 2003a)

555 Some non-stiff methods

The following method, for which c = [13,23, 1] , has order 2:

10 0 0 1 1130 11901

5 5

12 0 1 2360 4575

3 −29 12

an enhanced order of 3, but of course the stage order is only 2

The next method, with c = [14,12,34, 1] , has order 3:

364

5 28

364

5 28

For this method, possessing property F, β1=12, β2= 161 0= 0 The 3× 3

matrix ˙V is chosen so that δ(P −1 V P ) = 0, where˙

556 Some stiff methods

The first example, with λ = 14 and c = [14,12,34, 1] , has order 3:

283675 747648 312449

23364 −4525

396

1 36 1

4 1 −650

531

121459 46728

130127 124608

4z)4 ,

which makes the method L-stable

The second example has order 4 and an abscissa vector [ 1 3

4 1 4 1

4008881 197549232

58327 27726208

1 2 1 4

323 620736

65 11712

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 Nordsieckimplementations, 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 truncationerrors, for both the current method and for a method under consideration forsucceeding 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] , where D(r) = diag(1, r, r2, r3) This means that the V matrix

which determines stable behaviour for non-stiff problems, becomes effectively

7r2 −1

28r2 4

power-r2(1− r)/7 must lie in [−1, 1], so that r ∈ [0, r  ], where r  ≈ 2.310852163

is a zero of r3 = r2+ 7 For a product V (r n) V (r n −1)· · ·  V (r1), the non-zeroeigenvalue is %n

While this is a very mild restriction on r values for this method, the

corresponding restriction may be more severe for other methods For example,

for the scaled value of V given by (556b) the maximum permitted value of r

is approximately 1.725419906.

Whatever restriction needs to be imposed on r for stability, we may wish

to avoid even this restriction We can do this using a modification to simpleNordsieck scaling By Taylor expansion we find

40

21 −2

3 0 3221 17 −1

28(

to any row of the combined matrices [B |V ] without decreasing the order below

3 In the scale and modify procedure we can, after effectively scaling [B |V ] by D(r), modify the result by adding (1 − r2)d to the third row and 4(1 − r3)d

to the fourth row Expressed another way, write

558 Scale and modify for error estimation

Consider first the constant stepsize case and assume that, after many steps,

there is an accumulated error in each of the input components to step n If y(x) is the particular trajectory defined by y(x n −1 ) = y [n −1]

1 , then write theremaining input values as

y [n −1]

i = h i −1 y (i −1) (x

n −1) i −1 h p+1 y (p+1) (x n −1 ) + O(h p+2 ),

i = 2, 3, , p + 1. (558a)After a single step, the principal output will have acquired a truncation error

so that its value becomes y(x n) 0hp+1 y (p+1) (x n ) + O(h p+2), where

so that substitution of (558a) and (558c) into (558d), followed by Taylor

expansion about x n −1, gives the result

of (558b), as providing an estimate of the local error in a step, depends on

approximations to h p+1 y (p+1) (x n) are needed for stepsize control purposes

and, if these approximations are based on values of both hF and y [n −1], then

of h p+2 y (p+2) (x n) are needed as a basis for dynamically deciding when anorder increase is appropriate It was shown in Butcher and Podhaisky (2006)

that, at least for methods possessing property F, estimation of both h p+1 y (p+1) and h p+2 y (p+2)

In Subsection 557 we considered the method (555b) from the point of view

in a variable h regime, it is only necessary to add to the scaled and modified outputs y3[n] and y4[n], appropriate multiples of−hF1 + 3hF2− 3hF3 + hF4

Exercises 55 55.1 Show that the method given by (555a) has order 2, and that the stages

are also accurate to this order

55.2 Find the stability matrix of the method (555a), and show that it has

two zero eigenvalues

55.3 Show that the method given by (556a) has order 3, and that the stages

are also accurate to this order

55.4 Find the stability matrix of the method (556a), and show that it has

two zero eigenvalues

55.5 Show that (556a) is L-stable.

55.6 Show that the (i, j) element of Ψ −1 is equal to the coefficient of w i −1 z j −1

in the power series expansion about z = 0 of α(z)/(1 − (λ + w)z).

Alexander R (1977) Diagonally implicit Runge–Kutta methods for stiff ODEs

SIAM J Numer Anal., 14, 1006–1021.

Axelsson O (1969) A class of A-stable methods BIT, 9, 185–199.

Axelsson O (1972) A note on class of strongly A-stable methods BIT, 12, 1–4.

Barton D., Willers I M and Zahar R V M (1971) The automatic solution of

systems of ordinary differential equations by the method of Taylor series Comput.

J., 14, 243–248.

Bashforth F and Adams J C (1883) An Attempt to Test the Theories of Capillary

Action by Comparing the Theoretical and Measured Forms of Drops of Fluid, with

an Explanation of the Method of Integration Employed in Constructing the Tables which Give the Theoretical Forms of Such Drops Cambridge University Press,

Cambridge

Brenan K E., Campbell S L and Petzold L R (1989) Numerical Solution of

Initial-Value Problems in Differential-Algebraic Equations North-Holland, New York.

Brouder C (2000) Runge–Kutta methods and renormalization Eur Phys J C.,

12, 521–534

Burrage K (1978) A special family of Runge–Kutta methods for solving stiff

differential equations BIT, 18, 22–41.

Burrage K and Butcher J C (1980) Non-linear stability of a general class of

differential equation methods BIT, 20, 185–203.

Burrage K., Butcher J C and Chipman F H (1980) An implementation of

singly-implicit Runge–Kutta methods BIT, 20, 326–340.

Butcher J C (1963) Coefficients for the study of Runge–Kutta integration processes

J Austral Math Soc., 3, 185–201.

Butcher J C (1963a) On the integration processes of A Huˇta J Austral Math.

Soc., 3, 202–206.

Butcher J C (1965) A modified multistep method for the numerical integration of

ordinary differential equations J Assoc Comput Mach., 12, 124–135.

Butcher J C (1965a) On the attainable order of Runge–Kutta methods Math.

Comp., 19, 408–417.

Butcher J C (1966) On the convergence of numerical solutions to ordinary

differential equations Math Comp., 20, 1–10.

Butcher J C (1972) An algebraic theory of integration methods Math Comp., 26,

79–106

Butcher J C (1975) A 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

Butcher J C (1979) A transformed implicit Runge–Kutta method J Assoc.

Comput Mach., 26, 731–738.

Butcher J C (1985) The nonexistence of ten-stage eighth order explicit Runge–

Kutta methods BIT, 25, 521–540.

Butcher J C (1987) The Numerical Analysis of Ordinary Differential Equations,

Runge–Kutta and General Linear Methods John Wiley & Sons Ltd, Chichester.

Butcher J C (1987a) The equivalence of algebraic stability and AN-stability BIT,

Butcher J C (2002) The A-stability of methods with Pad´e and generalized Pad´e

stability functions Numer Algorithms, 31, 47–58.

Butcher J C (2006) General linear methods Acta Numerica, 15, 157–256.

Butcher J C (2008) Order and stability of generalized Pad´e approximations Appl.

Numer Math (to appear).

Butcher J C and Cash J R (1990) Towards efficient Runge–Kutta methods for

stiff systems SIAM J Numer Anal., 27, 753–761.

Butcher J C and Chartier P (1997) A generalization of singly-implicit Runge–

Kutta methods Appl Numer Math., 24, 343–350.

Butcher J C and Chipman F H (1992) Generalized Pad´e approximations to the

exponential function BIT, 32, 118–130.

Butcher J C and Hill A T (2006) Linear multistep methods as irreducible general

linear methods BIT, 46, 5–19.

Butcher J C and Jackiewicz Z (1996) Construction of diagonally implicit general

linear methods of type 1 and 2 for ordinary differential equations Appl Numer.

Math., 21, 385–415.

Butcher J C and Jackiewicz Z (1998) Construction of high order diagonally implicit

multistage integration methods for ordinary differential equations Appl Numer.

Math., 27, 1–12.

Butcher J C and Jackiewicz Z (2003) A new approach to error estimation for

general linear methods Numer Math., 95, 487–502.

Butcher J C and Moir N (2003) Experiments with a new fifth order method

Numer Algorithms, 33, 137–151

Butcher J C and Podhaisky H (2006) On error estimation in general linear methods

for stiff ODEs Appl Numer Math., 56, 345–357.

Butcher J C and Rattenbury N (2005) ARK methods for stiff problems Appl.

Numer Math., 53, 165–181

Butcher J C and Wright W M (2003) A transformation relating explicit and

diagonally-implicit general linear methods Appl Numer Math., 44, 313–327.

Butcher J C and Wright W M (2003a) The construction of practical general linear

methods BIT, 43, 695–721.

Butcher J C and Wright W M (2006) Applications of doubly companion matrices

Appl Numer Math., 56, 358–373.

15, 358–361

Butcher J C (1977) On A-stable implicit Runge–Kutta methods BIT, 17, 375–378.

Numerical Methods for Ordinary Differential Equations, ... class="text_page_counter">Trang 14< /span>

Alexander R (1977) Diagonally implicit Runge–Kutta methods for stiff ODEs

SIAM J Numer Anal., 14, 1006–1021.... method for the numerical integration of

ordinary differential equations J Assoc Comput Mach., 12, 124–135.

Butcher J C (1965a) On the attainable order of Runge–Kutta methods