Numerical Mathematics

Inversion of Spherical Means and the Wave Equation in Even Dimensions

“…and setting (B G) i := 0 for x i ∈ D. It is well known [15] that both linear interpolation in r and the trapezoidal rule in ϕ are second order approximations and therefore |(S x B − B S ϕ,r )G| ∞ ≤ C 3 max h 2 r , h 2 ϕ for some constant C 3 . In the numerical implementation, the coefficients in (54), are pre-computed and stored.…”

Iterative Methods for Stokes/Darcy Coupling

“…In fact, if we apply the PCG method which embeds the choice of the optimal acceleration parameter (see, e.g., Quarteroni et al [2000] p. 150), the iterative algorithm converges, but the optimal properties of the preconditioners are lost, since the number of iterations depends on the mesh parameter h, as reported in Table 2.…”

De3: Yet Another High-Order Integration Scheme for Linear Structural Dynamics

“…It also plays a role on the accuracy of the scheme which can be demonstrated to be fourth-order for β = 0 (numerically conservative scheme) or third-order for β ∈ (0, 1] (numerically dissipative scheme). Note that high-order accuracy requires that the integrals I 1 and I 2 be computed with at least third degree exactness, as is for instance offered by the Simpson-Cavalieri rule [8]. In such case, after completing the algebra relative to the definition of the error metrics introduced in [2], it can be shown that the local truncation errors for the free and forced responses are asymptotic to…”