Convolution Theorems for Quaternion Fourier Transform: Properties and Applications

General convolution theorems for two-dimensional quaternion Fourier transforms (QFTs) are presented. It is shown that these theorems are valid not only for real-valued functions but also for quaternion-valued functions. We describe some useful properties of generalized convolutions and compare them with the convolution theorems of the classical Fourier transform. We finally apply the obtained results to study hypoellipticity and to solve the heat equation in quaternion algebra framework

Convolution Theorems for Clifford Fourier Transform and Properties

The non-commutativity of the Clifford multiplication gives different aspects from the classical Fourier analysis.We establish main properties of convolution theorems for the Clifford Fourier transform. Some properties of these generalized convolutionsare extensions of the corresponding convolution theorems of the classical Fourier transform.DOI : http://dx.doi.org/10.22342/jims.20.2.143.125-14

THREE-DIMENSIONAL ADAPTIVE CENTRAL SCHEMES ON UNSTRUCTURED STAGGERED GRIDS ∗

Abstract. We present an explicit second-order finite volume generalization of the onedimensional (1D) Nessyahu–Tadmor schemes for hyperbolic equations on adaptive unstructured tetrahedral grids. The nonoscillatory central difference scheme of Nessyahu and Tadmor, in which the resolution of the Riemann problem at the cell interfaces is bypassed thanks to the use of the staggered Lax–Friedrichs scheme, is extended here to a two-steps scheme. In order to reduce artificial viscosity, we start with an adaptively refined primal grid in three dimensions (3D), where the theoretical a posteriori result of the first-order scheme is used to derive appropriate refinement indicators. We apply those methods to solve Euler’s equations. Numerical experimental tests on classical problems are obtained by our method and by the computational fluid dynamics software Fluent. These tests include results for the 3D Euler system (shock tube problem) and flow around an NACA0012 airfoil. Key words. 3D adaptive central schemes, unstructured staggered grid mesh adaptation, finite volume method