Partial Differential Equations - Modelling and ... - ResearchGate
Partial Differential Equations - Modelling and ... - ResearchGate Partial Differential Equations - Modelling and ... - ResearchGate
90 J.Ch. Gilbert and P. Joly 4.5 4 4.5 3.5 4 3 3.5 2.5 3 2 2.5 2 1.5 1.5 1 1 0.5 0.5 0 a 1,1 = 16 0 a 2,2 = 60.56 −0.5 0 2 4 6 8 10 12 14 16 −0.5 0 10 20 30 40 50 60 70 4.5 4.5 4 4 3.5 3.5 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 a 3,3 = 75.06 0 a 4,4 = 153.8 −0.5 0 10 20 30 40 50 60 70 80 −0.5 0 20 40 60 80 100 120 140 160 4.5 4.5 4 4 3.5 3.5 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 a 5,5 = 175.84 0 a 6,6 = 287.61 −0.5 0 20 40 60 80 100 120 140 160 180 −0.5 0 50 100 150 200 250 300 4.5 4.5 4 4 3.5 3.5 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 a 7,7 = 317.90 0 a 8,8 = 462.27 −0.5 0 50 100 150 200 250 300 350 −0.5 0 50 100 150 200 250 300 350 400 450 500 Fig. 4. The polynomials Q m = ψ m(0) (dashed curves) and the optimal polynomials ψ m(R m,m) for m =1,...,8 (solid curves)
Optimal Higher Order Time Discretizations 91 Table 2. Asymptotic behaviour of the diagonal schemes m ∆t m,m C m,m(T ) ∆t m,0 C 1,0(T ) 2α m,m m 2 π 2 1 2.00 1.00 3.24 2 3.89 1.03 3.07 3 4.33 1.39 1.69 4 6.20 1.29 1.95 5 6.63 1.51 1.43 6 8.48 1.42 1.62 7 8.91 1.57 1.31 8 10.75 1.49 1.46 ∞ 1.80 1.00 This conjecture is explored numerically in the third column of Table 2. Note that it does not distinguish between even and odd values of m, at least asymptotically. However, looking at the α m,m ’s on the diagonal of Table 1, it appears that the even values of k = m look more interesting than the odd ones. 5 Conclusion In this paper, we have analyzed the stability of higher order time discretization schemes for second order hyperbolic problems based on the modified equation approach. We have in particular proven that the upper bound for the time step (the CFL limit) remains uniformly bounded for large m (2m is the order of the scheme). On the basis of this information, we have proposed the construction of new schemes that are seen as modifications of the previous ones and are designed in order to optimize the CFL condition: this is formulated as an optimization problem in a space of polynomials of given degree. Despite some unpleasant properties (the objective function is non-convex and even discontinuous at the solution!), this problem can be fully analyzed. In particular, we prove the existence and uniqueness of the solution and give necessary and sufficient conditions of optimality. These conditions are exploited to design an algorithm for the effective numerical solution of the optimization problem. The obtained results are more than satisfactory with respect to our original objective. They suggest some conjectures that would mean that we would be able to produce schemes of arbitrary high order in time and whose computational cost would be almost independent of the order. Of course, this is a preliminary work and much has still to be done, including the following items:
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90 J.Ch. Gilbert <strong>and</strong> P. Joly<br />
4.5<br />
4<br />
4.5<br />
3.5<br />
4<br />
3<br />
3.5<br />
2.5<br />
3<br />
2<br />
2.5<br />
2<br />
1.5<br />
1.5<br />
1<br />
1<br />
0.5<br />
0.5<br />
0<br />
a 1,1 = 16<br />
0<br />
a 2,2 = 60.56<br />
−0.5<br />
0 2 4 6 8 10 12 14 16<br />
−0.5<br />
0 10 20 30 40 50 60 70<br />
4.5<br />
4.5<br />
4<br />
4<br />
3.5<br />
3.5<br />
3<br />
3<br />
2.5<br />
2.5<br />
2<br />
2<br />
1.5<br />
1.5<br />
1<br />
1<br />
0.5<br />
0.5<br />
0<br />
a 3,3 = 75.06<br />
0<br />
a 4,4 = 153.8<br />
−0.5<br />
0 10 20 30 40 50 60 70 80<br />
−0.5<br />
0 20 40 60 80 100 120 140 160<br />
4.5<br />
4.5<br />
4<br />
4<br />
3.5<br />
3.5<br />
3<br />
3<br />
2.5<br />
2.5<br />
2<br />
2<br />
1.5<br />
1.5<br />
1<br />
1<br />
0.5<br />
0.5<br />
0<br />
a 5,5 = 175.84<br />
0<br />
a 6,6 = 287.61<br />
−0.5<br />
0 20 40 60 80 100 120 140 160 180<br />
−0.5<br />
0 50 100 150 200 250 300<br />
4.5<br />
4.5<br />
4<br />
4<br />
3.5<br />
3.5<br />
3<br />
3<br />
2.5<br />
2.5<br />
2<br />
2<br />
1.5<br />
1.5<br />
1<br />
1<br />
0.5<br />
0.5<br />
0<br />
a 7,7 = 317.90<br />
0<br />
a 8,8 = 462.27<br />
−0.5<br />
0 50 100 150 200 250 300 350<br />
−0.5<br />
0 50 100 150 200 250 300 350 400 450 500<br />
Fig. 4. The polynomials Q m = ψ m(0) (dashed curves) <strong>and</strong> the optimal polynomials<br />
ψ m(R m,m) for m =1,...,8 (solid curves)
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