Monotonicity and Boundedness in general Runge-Kutta methods
Ferracina, L.Citation
Ferracina, L. (2005, September 6). Monotonicity and Boundedness in general Runge-Kutta methods. Retrieved from https://hdl.handle.net/1887/3295
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behorende bij het proefschrift
Monotonicity and Boundedness in
general Runge-Kutta methods
van Luca Ferracina
In the following propositions we deal with the numerical solution of initial value problems for sytems of ordinary differential equations that can be written in the form
d
dtU (t) = F (U (t)) (t ≥ 0), U(0) = u0. (1)
We focus on the (irreducible) Runge-Kutta method (2) where, given an approximation un−1 of U (tn−1), a new approximation un of U (tn−1+ ∆t) is computed by the relations
yi = un−1+ ∆t s X j=1 κijF (yj) (1 ≤ i ≤ s + 1), un = ys+1. (2)
We identify the Runge-Kutta method with the (s + 1) × s matrix K = (κij) and we denote
with K0 the s × s submatrix K0 = (κij), 1 ≤ i ≤ j ≤ s. We are interested in coefficients c
such that kv + τ0F (v)k ≤ kvk (∀ v ∈ V) 0 < ∆t ≤ c · τ0 ⇒ kunk ≤ kun−1k. (3)
Let L = (λij) be any (s + 1) × s matrix with submatrix L0 = (λij), 1 ≤ i ≤ j ≤ s such
that L ≥ 0, Les ≤ es+1 and I − L0 is invertible – here, and in the following, I denotes the
s × s identity matrix and em ∈ Rm stands for the column vector with all components equal
to 1 (for m = s, s + 1). Define the (s + 1) × s matrix M = (µij) by M = K − LK0 and
consider the process yi= 1 − s X j=1 λij ! un−1+ s X j=1 [λijyj + ∆t · µijF (yj)] (1 ≤ i ≤ s + 1), un= ys+1. (4)
We define the following coefficient c(L, M ) = min{γij : 1 ≤ i ≤ s + 1, 1 ≤ j ≤ s}, γij= λij/µij if µij > 0, ∞ if µij = 0, 0 if µij < 0.
1. Process (4) is a useful representation of process (2). The following two statements are valid.
(i) Method (2) and process (4) are equivalent.
(ii) Let c be equal to c(L, M ) defined above. Then implication (3) holds whenever V is a real vector space with seminorm k.k, and un, un−1 are related to each other as in (4).
See Chapters I and II of this thesis.
Given a Runge-Kutta method K, consider, for real γ, the following conditions: (I + γK0) is invertible, γK(I + γK0)−1 ≥ 0, γK(I + γK0)−1es≤ es+1.
(5)
We define the following coefficient
R(K) = sup{γ : γ ≥ 0 and (5) holds}. 2. The largest c guaranteeing (3) for methods (2).
Let c be given with 0 < c ≤ ∞. Then (I) and (II) are equivalent: (I) c ≤ R(K),
(II) implication (3) holds whenever V is a real vector space with seminorm k.k, and un,
un−1 are related to each other as in (2).
See Chapters I and II of this thesis.
3. Optimal (L, M ) representations.
For any Runge-Kutta method K there exist an (L, M ) representation with c(L, M ) = R(K).
See Chapter II of this thesis.
4. The optimal (L, M ) representation is not unique.
5. Optimal Runge -Kutta methods.
Let C be a given class of Runge-Kutta methods K such that c∗ = max{R(K) : K ∈ C }
exists and is finite. We denote by ¯C the set of all (L, M ) representaions of methods K ∈ C . Then the following two statements are valid.
(i ) The maximum of γ, specified in the following two procedures, exists and equals c∗.
(ii ) The first procedure is, from a practical point of view, to be preferred over the second one.
Procedure 1 maximize γ, subject to: γ satisfies (5) and K ∈ C . Procedure 2 maximize γ, subject to: L − γ M ≥ 0 and (L, M) ∈ ¯C.
See Chapter III of this thesis.
6. Completing results in the literature.
In the literature, optimal (w.r.t. R(K)) explicit Runge-Kutta methods, with s stages and order of accuracy at least p, are available with 1 ≤ p ≤ 4 and p ≤ s ≤ 9, except the case (s, p) = (9, 4). It can be shown that the missing optimal method K has R(K) = 4.9142 (rounded to 5 decimal digits).
7. Boundedness.
Statement 2 can be generalized so as to become valid also when (3) is replaced by the following implication
kv + τ0F (v)k ≤ (1 + α0τ0)kvk + β0τ0 (∀ v ∈ V)
0 < ∆t ≤ c · τ0
⇒ kunk ≤ (1 + α∆t)kun−1k + β∆t.
See Chapter IV of this thesis.
8. TVD does not avoid oscillations.
When dealing with numerical solutions of IVPs for ODEs (and PDEs), one should keep in mind the following remark.
“(...) some people believe that the TVD property (i.e. kunkT V ≤ kun−1kT V with k.kT V total
variation seminorm) completely eliminates all spurious oscillations for all (∆x and) ∆t. It does not. In fact, the TVD condition may allow large spurious oscillations (...)”
In the following two statements we denote by Ss,p the class of all singly-diagonally-implicit
s-stage Runge-Kutta methods K = (κij), with order of accuracy at least p and with
all κij ≥ 0, κii> 0.
9. Upper bound for the order of accuracy. There are no methods, with R(K) > 0, in Ss,p if p > 4.
10. Optimal method in S3,4.
Consider the following Runge-Kutta method
K3,4 = 1+ξ 2 0 0 −ξ 2 1+ξ 2 0 1 + ξ −1 − 2ξ 1+ξ2 1 6ξ2 1 − 1 3ξ2 1 6ξ2 with ξ = −√2 3cos( 5π 18).
Then K3,4 ∈ S3,4, and for any other K ∈ S3,4 we have R(K) < R(K3,4) = 2
1 + ξ ξ2
− ξ − 1. 11. A model for studying the dispersion in the Venice Lagoon
This is a mesh with 1967 nodes and 3423 triangular ele-ments modelling the Venice Lagoon. Discretizing in space the advection-diffusion equation
∂u
∂t + v · ∇u = ∇ · (K · ∇u) + s
with the finite element method (linear triangular elements), one obtains a semi-discrete system of ordinary differential equations that can be written as
Md
dtU (t) + N(t)U (t) + l(t) = 0. Consider the simple time-discretization (fully-discrete system)
Mun+1− un
∆t + Nnun+ θ[Nn+1un+1− Nnun] + ln+ θ(ln+1− ln) = 0. We then obtain a linear system (in Rm, m = 1967), of the form
Aun+1 = b,
