A Direct Spectral Method for Wave Propagation Modelling

Abstract eng:
Most computational methods for investigation of physical phenomena are based on numerically solving a related system of algebraic equations. Such a system is classically obtained by discretization of the space-time continuum where the physical laws have been formulated either in differential or in integral form. Finite Difference, Finite Element, Finite Volume methods are examples of such an approach. It has been recently shown that a direct algebraic formulation (DAF) is possible in such a way that the solving system can be easily obtained by directly writing the physical laws in a discrete form (Cell Method), thus totally avoiding the use of any differential formulation. It is a PDE-free approach which leads naturally to a system of equations to be solved numerically and which is based on the use of global physical variables associated with discrete spatial and temporal elements, such as points, lines, surfaces, volumes, time instants and intervals. Balance and circuital laws can be expressed exactly in a global form. In order to carry out such a formulation it is necessary to partition the working region into spatial-temporal cells, by means of a cell complex and its dual complex. In the case of the wave propagation the discrete operator is then built directly by using the corresponding balance and conservation laws with the global variables related to the mesh elements. The approach leads to computational meshes that can be structured or unstructured with elementary cells having, a priori, any shape. A computational scheme can be easily derived which is geometrically flexible, staggered and accurate to second order both in space and in time. High order schemes are needed for more accurate and efficient wave propagation modelling. In the present work we describe an improved numerical method based on both DAF and spectral approaches. A direct spectral method (DSM) is derived in which primal and dual cells are built using Chebyshev collocation points and the algebraic solver is directly derived on such cells. Numerical results for 2D acoustic wave propagation will be shown that assess the feasibility and the spectral accuracy of the present approach.

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