Abstract
AbstractA new approach to building explicit time‐marching stencil computation schemes for the transient two‐dimensional acoustic wave equation is implemented. It is based on using Poisson's formula and its three time level modification combined with polynomial stencil interpolation of the solution at each time‐step and exact integration. The time‐stepping algorithm consists of two explicit stencil computation procedures: a first time‐step procedure incorporating the initial conditions and a two‐step scheme for the second and next time‐steps. Three particular explicit stencil schemes (with 5, 9, and 13 space points) are constructed using this approach. Their stability regions are presented. All of the obtained first time‐step computation expressions are different from those used in conventional finite‐difference methods. Accuracy advantages of the new schemes in comparison with conventional finite‐difference schemes are demonstrated by simulation using an exact benchmark solution.
Highlights
Stencil computations are widely implemented in many numerical algorithms that involve structured grids, including finite-difference techniques in such diverse areas as acoustics, fluid dynamics, solid mechanics and electromagnetism
Among the explicit schemes for the transient 2D wave equation the main attention in the literature has been given to two-step schemes (which operate over three time levels tk+1 = (k + 1)τ, tk = kτ and tk−1 = (k − 1)τ where τ is a fixed time increment)
This paper focuses on building explicit time-stepping stencil computation schemes for the transient 2D acoustic wave equation without using finitedifference methods
Summary
Stencil computations are widely implemented in many numerical algorithms that involve structured grids, including finite-difference techniques in such diverse areas as acoustics, fluid dynamics, solid mechanics and electromagnetism. Among the explicit schemes for the transient 2D wave equation the main attention in the literature has been given to two-step schemes (which operate over three time levels tk+1 = (k + 1)τ, tk = kτ and tk−1 = (k − 1)τ where τ is a fixed time increment). This paper focuses on building explicit time-stepping stencil computation schemes for the transient 2D acoustic (scalar) wave equation without using finitedifference methods. The approach implemented here is based on using a famous integral representation formula (Poisson’s formula) that provides the exact solution of the initial-value problem for the transient 2D scalar wave equation at any time point through the initial conditions. It is shown by simulation that the derived numerical schemes can significantly improve accuracy of stencil computations in comparison with conventional approaches
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