By Arieh Iserles

ISBN-10: 1107010861

ISBN-13: 9781107010864

Acta Numerica is an annual booklet containing invited survey papers via best researchers in numerical arithmetic and medical computing. The papers current overviews of contemporary advancements of their zone and supply 'state of the artwork' options and research.

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**Sample text**

2. Spatial symmetries and high-order discretizations In this subsection we will discuss novel techniques for spatial discretizations based on reﬂection groups and multivariate Chebyshev polynomials. Given high-order time integration methods, such as Lie group methods, it is desirable to also use high-order discretizations in space. We will see that it is important to employ discretizations respecting spatial symmetries, both for the quality of the discretization error and also for eﬃciency of linear algebra computations such as matrix exponentials, eigenvalue computations and solution of linear systems.

As orbit representatives, we may pick S = {1, 7, 10}. The action of the symmetry group is free on the orbit O1 = {1, 2, 3, 4, 5, 6}, while the points in the orbit O7 = {7, 8, 9} have isotropy subgroups of size 2, and ﬁnally O10 = {10} has isotropy of size 6. The operator L is discretized as a matrix A ∈ C10×10 satisfying the equivariances Aig,jg = Ai,j for g ∈ {α, β} and i, j ∈ S. , A1,6 = A3,2 = A5,4 = A4,5 = A2,3 = A6,1 . 14)). 19). The transformed maˆ 1 ) ∈ Cm×m and A(ρ ˆ 2 ) ∈ Cm×m ⊗C2×2 ˆ 0 ), A(ρ trix Aˆ has three blocks, A(ρ 2m×2m , where m = 3 is the number of orbits.

In particular, if φ∗ F = F , we say that φ is a symmetry of the vector ﬁeld, and in that case φ maps solution curves to other solution curves of the same equation. For numerical integrators it is in general impossible to satisfy equivariance with respect to arbitrary diﬀeomorphisms, since this would imply an analytically correct solution. ) The equivariance group of a numerical scheme is the largest group of diffeomorphisms under which the numerical solutions transform equivariantly. It is known that the equivariance group of classical Runge–Kutta methods is the group of all aﬃne linear transformations of Rn .