Numerical Simulation of Time-Harmonic Waves in Inhomogeneous Media using Compact High Order Schemes

Abstract

This paper builds on the outcomes of a state-of-the-art review of research for the European research project MULTAGRI, with a particular focus on the conceptualisation of Multifunctional Agriculture (MFA) from a 'supply side' perspective. It argues that a consistent and satisfactory conceptual framework for studying the supply of MFA at farm and territorial level is still largely lacking and that, when consistently applied, the recognition of the multifunctionality of agriculture fundamentally challenges the ways in which agricultural activities are observed, analysed and their effects assessed. This is elaborated by discussing five 'critical questions' for studying the supply of MFA:(1) what are relevant functions of agriculture? (2) how to define agricultural activities? (3) how to identify relevant farm/firm categories? (4) how to study the role of changing rural identities? and (5) what is the role of new territorial and institutional arrangements? (2008) 'Multifunctionality of agricultural activities, changing rural identities and new institutional arrangements', Int.

The Method of Difference Potentials for the Helmholtz Equation Using Compact High Order Schemes

“…Higher order accuracy can be achieved without expanding the stencil in schemes known as Collatz "Mehrstellen" [10], equation-based and related compact schemes [2,3,5,6,20,31,47,48,51] and Trefftz-FLAME [54]. Such schemes rely on a targeted approximation of the class of solutions rather than of a much broader class of generic sufficiently smooth functions.…”

“…In [6], we have extended the analysis to the case of a more general Helmholtz equation, with variable coefficients under the derivatives in the Laplacian-like part.…”

“…Extension of scheme (10) to three space dimensions is straightforward, see [6]. In 3D, the scheme will employ a 3 × 3 × 3 stencil on the left-hand side and a seven-node central difference stencil on the right-hand side.…”

High Order Compact Scheme and the Moving Mesh Method for Helmholtz Equation with Variable Wave Numbers

“…Starting from the noticeable work [1], a family of high-order compact schemes are derived for variety of Helmholtz equations based on the differencing of governing equation to eliminate higher order derivatives in the discretization error, such as, the compact finite difference scheme for one dimension problems [2], the compact fourth order accuracy finite difference approximation for two and three dimension problems on the Cartesian coordinate [1,3] and polar coordinates [4,5]. Other papers have extended the HOC scheme to Helmholtz equation with non-homogeneous materials [6], with nonstandard boundary conditions [7] and with singular solutions [8]. Recently, more accurate compact sixth order methods are considered for solving the Helmholtz equation with a constant wave number [9,10,11] and a variable wave number [12].…”

On the Derivation of Highest-Order Compact Finite Difference Schemes for the One- and Two-Dimensional Poisson Equation with Dirichlet Boundary Conditions

“…For higher-dimensional problems, compact 19-point and 25-point standard CDS of fourth-order local accuracy were presented for the three-and four-dimensional problem, respectively [4,45,50,56]. Similarly, high-order scheme studies have been performed on related problems such as the Poisson equation with Neumann boundary conditions [6], Laplace equation [52,2], Helmholtz equation [2,34,3,25,51,49,35,41,40,7], biharmonic equation [26,27,46], convection-diffusion equations [8,23,24,32,31,29,30,28,48,47,22,19] with various boundary conditions, and stream function vorticity equations [42,43,44,57].…”