Partial Differential Equations - Modelling and ... - ResearchGate

Optimal Higher Order Time Discretizations 85
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
−0.5
0 5 10 15 20 25 30 35 40 45 50
Fig. 2. Checking the sufficient condition of optimality for m =4andk =1
has a tangent point at τ 1 ≃ 9.88, but is not optimal since the value of the
polynomial at this point does not satisfy [ψ 4 (R 4,1 )](τ 1 )+[ψ 4 (R 4,1 )](τ 0 )=4
(for this polynomial τ 0 ≃ 34.22).
⊓⊔
4 Computational Issues
4.1 Algorithm Based on the Parametrization by the Tangent
Points
In the numerical results discussed below, the optimal polynomial is searched
by its k alternate tangent points (τ j ) 1≤j≤k , with τ 1 >τ 2 > ···>τ k , whose existence
is ensured by Theorem 4. These points are determined in the following
manner. For τ =(τ 1 ,...,τ k ), let R(τ) be the polynomial in P k−1 satisfying
ψ m (R(τ)) = v ∈ R k ,
in which the components of v take alternatively the values 0 and 4. Whether
one has to impose v 1 =0orv 1 = 4 is further discussed below. The coefficients
r =(r 0 ··· r k−1 ) T of R(τ) are uniquely determined by the equation above,
which can also be written
⎛
τ1 m+1 ··· τ m+k ⎞ ⎛
⎞
1
[ψ m (0)] (τ 1 )
⎜
⎝
.
⎟ ⎜
. ⎠ r = v −
⎝
⎟
. ⎠ . (30)
τ m+1
k
··· τ m+k
[ψ
k
m (0)] (τ k )
Next, let us introduce the function F : τ ∈ R k ↦→ F (τ) ∈ R k ,wherethe
components of F (τ) are the derivatives of the polynomial ψ m (R(τ)) at the
τ j ’s: