The collocation method for ODEs: an introduction

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1 Collocation Methods for Volterra Integral Related Functional Differential The collocation method for ODEs: an introduction A collocation solution u h to a functional equation for example an ordinary differential equation or a Volterra integral equation) on an interval I is an element from some finite-dimensional function space the collocation space) which satisfies the equation on an appropriate finite subset of points in I the set of collocation points) whose cardinality essentially matches the dimension of the collocation space. If initial or boundary) conditions are present then u h will usually be required to fulfil these conditions, too. The use of polynomial or piecewise polynomial collocation spaces for the approximate solution of boundary-value problems has its origin in the 930s. For initial-value problems in ordinary differential equations such collocation methods were first studied systematically in the late 960s: it was then shown that collocation in continuous piecewise polynomial spaces leads to an important class of implicit high-order) Runge Kutta methods.. Piecewise polynomial collocation for ODEs.. Collocation-based implicit Runge Kutta methods Consider the initial-value problem y t) = f t, yt)), t I := [0, T, y0) = y 0,..) assume that the Lipschitz-) continuous function f : I IR IR is such that..) possesses a unique solution y C I ) for all y 0. Let I h :={t n : 0 = t 0 < t <...<t N = T } be a given not necessarily uniform) mesh on I, set σ n := t n, t n+, σ n := [t n, t n+, with h n := t n+ t n n = 0,,...,N ). The quantity

2 Collocation Methods for Volterra Integral Related Functional Differential The collocation method for ODEs: an introduction h := max{h n : 0 n N } will be called the diameter of the mesh I h ; in the context of time-stepping we will also refer to h as the stepsize. Note that we have, in rigorous notation, t n = t N) n, σ n := σ N) n, h n = h N) n n = 0,,...,N ), h = h N). However, we will usually suppress this dependence on N, the number of subintervals corresponding to a given mesh I h, except occasionally in the convergence analyses where N, h = h N) 0, so that Nh N) remains uniformly bounded. The solution y of the initial-value problem..) will be approximated by an element u h of the piecewise polynomial space S 0) m I h):={v CI ): v σn π m 0 n N )},..) where π m denotes the space of all real) polynomials of degree not exceeding m. It is readily verified that S 0) m I h) is a linear space whose dimension is dim S 0) m I h) = Nm + a description of more general piecewise polynomial spaces will be given in Section..). This approximation u h will be found by collocation; that is, by requiring that u h satisfy the given differential equation on a given suitable finite subset X h of I, coincide with the exact solution y at the initial point t = 0. It is clear that the cardinality of X h, the set of collocation points, will have to be equal to Nm, the obvious choice of X h is to place m distinct collocation points in each of the N subintervals σ n. To be more precise, let X h be given by X h :={t = t n + c i h n : 0 c <...<c m 0 n N )}...3) For a given mesh I h, the collocation parameters {c i } completely determine X h. Its cardinality is X h { Nm if 0 < c <...<c = m or 0 c <...<c m < ), Nm ) + if 0 = c < c <...<c m = m ). The collocation solution u h S 0) m I h) for..) is defined by the collocation equation u h t) = f t, u ht)), t X h, u h 0) = y0) = y 0...) If u h corresponds to a set of collocation points with c = 0 c m = m ), it lies if it exists on I ) in the smoother space S m 0)I h) C I ) =: S m )I h)of dimension Nm ) + whenever the given function f in..) is continuous. This follows readily by considering the collocation equation..) at

3 Collocation Methods for Volterra Integral Related Functional Differential. Piecewise polynomial collocation for ODEs 3 t = t n + c m h n =: tn at t = t n + c h n =: t n + : taking the difference using the continuity of f leads to u h t n + ) u h t n ) = 0, n =,...,N, this is equivalent to u h being continuous at t = t n. In order to obtain more insight into this piecewise polynomial collocation method, to exhibit its recursive nature, we now derive the computational form of..). This will reveal that the collocation equation..) represents the stage equations of an m-stage continuous implicit Runge Kutta method for the initial-value problem..) compare also the original papers by Guillou Soulé 969), Wright 970), or the book by Hairer, Nørsett Wanner 993). Here, in subsequent chapters of the book, it will be convenient natural) to work with the local Lagrange basis representations of u h. Since u h σ n π m,wehave u h t n + vh n ) = L j v)y n, j, v 0,, Y n, j := u h t n + c j h n ),..5) where the polynomials L j v) := m v c k c j c k k j j =,...,m), denote the Lagrange fundamental polynomials with respect to the distinct) collocation parameters {c i }. Setting y n := u h t n ) β j v) := v 0 L j s)ds j =,...,m), we obtain from..5) the local representation of u h S m 0)I h)on σ n, namely u h t n + vh n ) = y n + h n β j v)y n, j, v [0,...6) The unknown derivative) approximations Y n,i i =,...,m) in..6) are defined by the solution of a system of generally nonlinear) algebraic equations obtained by setting t = t n,i := t n + c i h n in the collocation equation..) employing the local representations..5)..6). This system is ) Y n,i = f t n,i, y n + h n a i, j Y n, j, i =,...,m),..7) where we have defined a i, j := β j c i ).

4 Collocation Methods for Volterra Integral Related Functional Differential The collocation method for ODEs: an introduction We see that the equations..6)..7) define, as asserted above, a continuous implicit Runge Kutta CIRK) method for the initial-value problem..): its m stage values Y n,i are given by the solution of the nonlinear algebraic systems..7),..6) defines the approximation u h for each t σ n n = 0,,...,N ). This local representation may be viewed as the natural interpolant in π m on σ n for the data {t n, y n ), t n,i, Y n,i )i =,...,m)}. It thus follows that such a continuous implicit RK method contains an embedded classical discrete) m-stage implicit Runge Kutta method for..): it corresponds to..6) with v =, y n+ := u h t n + h n ) = y n + h n b j Y n, j n = 0,,...,N ),..8) with b j := β j ), the stage equations..7). If m if the collocation parameters {c i } are such that 0 = c < c <...<c m =, then t n, = t n implies Y n, = f t n, y n ), the CIRK method..6),..7) reduces to u h t n + vh n ) = y n + h n β v) f t n, y n ) + h n β j v)y n, j, v [0,, Y n,i = f t n,i, y n + h n a i, f t n, y n ) + h n Moreover, since c m =, we obtain Y n,m = f j= ) a i, j Y n, j j= t n+, y n + h n b f t n, y n ) + h n..9) i =,...,m). ) b j Y n, j. j=..0) Example.. u h S 0) I h)m = ), with c =: θ [0, : Since L v) β v) = v hence a, = θ b = ),..6) reduces to u h t n + vh n ) = y n + h n vy n,, v [0,, with Y n, defined by the solution of Y n, = f t n + θh n, y n + h n θy n, ). These equations may be combined into a single one by setting v = in the expression for u h t n + vh n ) solving for Y n, ); the resulting method is the

5 Collocation Methods for Volterra Integral Related Functional Differential. Piecewise polynomial collocation for ODEs 5 continuous θ-method for..), where implicitly defines y n+. u h t n + vh n ) = v)y n + vy n+, v [0,. y n+ = y n + h n f t n + θh n, θ)y n + θy n+ ) This family of continuous one-stage Runge Kutta methods contains the continuous implicit Euler method θ = ) the continuous implicit midpoint method θ = /). For θ = 0 we obtain the continuous explicit Euler method. Due to its importance in the time-stepping of spatially) semidiscretised parabolic PDEs or PVIDEs) we state the continuous implicit midpoint method for the linear ODE y t) = at)yt) + gt), t I, with a g in CI ). Setting θ = / we obtain y n+ = y n + h n at n + h n /)[y n + y n+ + gt n + h n /)n = 0,,...,N ), or, using the notation t n+/ := t n + h n /, h ) n at n+/) y n+ = + h ) n at n+/) y n + h n gt n+/ )...) Observe the difference between..) the continuous trapezoidal method: the latter corresponds to collocation in the space S 0) I h), with c = 0, c = being the Lobatto points; it is described in Example.. below m = ). Example.. u h S 0) I h)m = ), with 0 c < c : It follows from L v) = c v)/c c ), L v) = v c )/c c ) that β v) = vc v) c c ), β v) = vv c ) c c ). Hence, b = β ) = c )/c c )), b = β ) = c )/c c )). The resulting continuous two-stage Runge Kutta method thus reads: where u h t n + vh n ) = y n + h n {β v)y n, + β v)y n, }, v [0,, Y n,i = f t n,i, y n + h n {a i, Y n, + a i, Y n, }) i =, ).

6 Collocation Methods for Volterra Integral Related Functional Differential 6 The collocation method for ODEs: an introduction We present three important special cases: Gauss points c = 3 3)/6, c = 3 + 3)/6: We obtain β v) = v + 3 v))/, β v) = v 3 v))/, A := [ a i, j = [ , b = [ b b = The discrete version of this two-stage implicit Runge Kutta Gauss method of order ; cf. Section..3, Corollary..6) was introduced by Hammer Hollingsworth 955) generalised by Kuntzmann in 96 see Ceschino Kuntzmann 963) for details). Radau II points c = /3, c = : Here, we have β v) = 3v v)/, β v) = 3vv /3)/, [ 5 A =, b = 3 [ 3 This represents the continuous two-stage Radau IIA method. Lobatto points c = 0, c = = u h S ) I h)): The continuous weights are now hence β v) = v v)/, β v) = v /, [ 0 0 A =, b = This yields the continuous trapezoidal method: it can be written in the form [.. [. with u h t n + vh n ) = y n + h n v v)yn, + v Y n, ), v [0,, Y n, = f t n, y n ), Y n, = f t n+, y n + h n /){Y n, + Y n, }). See also Hammer Hollingsworth 955).)

7 Collocation Methods for Volterra Integral Related Functional Differential. Piecewise polynomial collocation for ODEs 7 For the linear ODE y t) = at)yt) + gt) the stage equation assumes the form h ) nat n+ ) Y n, = + h ) nat n ) at n+ )y n + h nat n+ ) gt n ) + g)t n+ ). Remark Other examples of discrete) RK methods based on collocation, including methods corresponding to the Radau I points c = 0, c = /3 when m = ), may be found for example in the books by Butcher 987, 003), Lambert 99), Hairer Wanner 996). There is an alternative way to formulate the above continuous implicit Runge Kutta method..6),..7). Setting U n,i := y n + h n a i, j Y n, j i =,...,m), we obtain the symmetric formulation u h t n + vh n ) = y n + h n β j v) f t n, j, U n, j ), v [0,,..) with U n,i = y n + h n a i, j f t n, j, U n, j ) i =,...,m)...3) Here, the unknown stage values U n,i represent aproximations to the solution y at the collocation points t n,i i =,...,m). For v =,..) yields the symmetric analogue of..8), y n+ = y n + h n b j f t n, j, U n, j );..) if c m = wehavey n+ = U n,m. For later reference, to introduce notation needed later, we also write down the above CIRK method..6),..7) for the linear initial-value problem y t) = at)yt), t I, y0) = y 0, where a CI ). Setting A := a i, j ) LIR m ),βv) := β v),...,β m v)) T IR m, Y n := Y n,,...,y n,m ) T IR m, the CIRK method can be written in the form u h t n + vh n ) = y n + h n β T v)y n, v [0,,..5) with Y n given by the solution of the linear algebraic system [I m h n A n Y n = diagat n,i ))e y n n = 0,,...,N )...6)

8 Collocation Methods for Volterra Integral Related Functional Differential 8 The collocation method for ODEs: an introduction Here, I m denotes the identity in LIR m ), A n := diagat n,i ))A, e :=,...,) T IR m. The derivation of the analogue of..5),..6) corresponding to the symmetric formulation..),..3) of the CIRK method is left as an exercise Exercise.0.). The classical conditions for the existence uniqueness of a solution y C I ) to the initial-value problem..) see, e.g. Hairer, Nørsett Wanner 993, Sections I.7 I.9) assure the existence uniqueness of the collocation solution u h S m 0)I h) to..) or its linear counterpart for all h := max n) h n in some interval 0, h), provided that f y is bounded or a g lie in CI ) when the ODE is y = at)y + gt)). In the latter case, the existence of such an h follows from the Neumann Lemma which states that I m h n A n ) is uniformly bounded for all sufficiently small h n > 0, so that h n A n < for some operator) matrix norm. We shall give the precise formulation of this result in in Chapter 3 Theorem 3..) for VIDEs which contains the version for ODEs as a special case. It is clear that not every implicit Runge Kutta method can be obtained by collocation as described above see, for example, Nørsett 980), Hairer, Nørsett Wanner 993)): a necessary condition is clearly that the parameters c i are distinct. The framework of perturbed collocation Nørsett 980), Nørsett Wanner 98); see also Section. below) encompasses all implicit) Runge Kutta methods. There is also an elegant connection between continuous Runge Kutta methods discontinuous collocation methods Hairer, Lubich Wanner 00, pp. 3 3)). The following result which can be found in Hairer, Nørsett Wanner 993, p. )) characterises those implicit Runge Kutta methods that are collocation-based. Theorem.. The m-stage implicit Runge Kutta method defined by..7)..8), with distinct parameters c i order at least m, can be obtained by collocation in S 0) m I h), as described above, if only if the relations hold. a i, j c ν j = cν i, ν =,...,m i =,...,m), ν The proof of this result is left as an exercise. Recall that a discrete) Runge Kutta method for..) is said to be of order p if yt ) y Ch p

9 Collocation Methods for Volterra Integral Related Functional Differential. Piecewise polynomial collocation for ODEs 9 for all sufficiently smooth f = f t, y) in..). The next section will reveal that the collocation solution u h S m 0)I h) to..) is of global order p m on I... Convergence global order on I Suppose that the collocation equation..) defines a unique collocation solution u h S m 0)I h) for all sufficiently small mesh diameters h 0, h). What are the optimal values of p ν pν ν = 0, ) in the global local) error estimates y ν) u ν) h := sup t I y ν) t) u ν) h t) C νh p ν..7) y ν) u ν) h h, := max t I h \{0} yν) t) u ν) h t) C νh p ν,..8) respectively? These values depend of course on the regularity of the solution y of the initial-value problem..). For arbitrarily regular y we will refer to the largest attainable p ν ν = 0, ) as the optimal) orders of global super-) convergence on the interval I )ofu h u h, respectively, the corresponding pν will be called the optimal) orders of local superconvergence at the mesh points I h \{0}) ofu h u h, provided p ν > p ν. In order to introduce the essential ideas underlying the answer to the above question regarding the optimal orders, we first present the result on global convergence for the linear initial-value problem Theorem.. Assume that y t) = at)yt) + gt), t I, y0) = y 0...9) a) the given functions in..9) satisfy a, g C m I ); b) the collocation solution u h S m 0)I h) for the initial-value problem..9) corresponding to the collocation points X h is defined by..5),..6); c) h > 0 is such that, for any h 0, h), each of the linear systems..6) has a unique solution. Then the estimates y u h := max yt) u h t) C 0 y m+) h m..0) t I y u h := sup y t) u h t) C y m+) h m,..) t I

10 Collocation Methods for Volterra Integral Related Functional Differential 0 The collocation method for ODEs: an introduction hold for h 0, h) any X h with 0 c <...<c m. The constants C ν depend on the collocation parameters {c i } but are independent of h, the exponent m of h cannot in general be replaced by m +. Proof Assumption a) implies that y C m+ I ) hence y C m I ). Thus we have, using Peano s Theorem Corollary.8. with d = m) for y on σ n, y t n + vh n ) = L j v)z n, j + h m n R) m+,n v), v [0,,..) with Z n, j := y t n, j ). The Peano remainder term Peano kernel are given by K m v, z) := R ) m+,n v) := { v z) + m m )! Integration of..) leads to where yt n + vh n ) = yt n ) + h n see also Exercise.0.3). 0 K m v, z)y m+) t n + zh n )dz,..3) R m+,n v) := k= L k v)c k z) m + }, v [0,. β j v)z n, j + h m+ n R m+,n v), v [0,, v 0 R ) m+,n s)ds..) Recalling the local representation..6) of the collocation solution u h on σ n, setting E n, j := Z n, j Y n, j, the collocation error e h := y u h on σ n may be written as e h t n + vh n ) = e h t n ) + h n while e h t n + vh n ) = β j v)e n, j + h m+ n R m+,n v), v [0,,..5) L j v)e n, j + h m n R) m+,n v), v 0,,..6)

Runge-Kutta and Collocation Methods Florian Landis

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