Abstract
In this paper, we devote ourselves to establishing the unconditionally stable and absolutely convergent optimized finite difference Crank-Nicolson iterative (OFDCNI) scheme containing very few degrees of freedom but holding sufficiently high accuracy for the two-dimensional (2D) Sobolev equation by means of the proper orthogonal decomposition (POD) technique, analyzing the stability and convergence of the OFDCNI solutions and using the numerical simulations to verify the feasibility and effectiveness of the OFDCNI scheme.
Highlights
For convenience and without loss of universality, we think about the following twodimensional ( D) Sobolev equation: ⎧ ⎪⎨ ∂u ∂t ε ∂ ∂u t γ u = f (x, y, t), (x, y, t) ∈ × (, T),⎪⎩uu((xx, y, y, t) = Q(x, y, t), ) = G(x, y),(x, y, t) ∈ ∂ (x, y) ∈, ×
The optimized finite difference Crank-Nicolson iterative (OFDCNI) scheme has fully second-order accuracy, is unconditionally stable and absolutely convergent, and is only built by the proper orthogonal decomposition (POD) basis constituted with the classical finite difference Crank-Nicolson (FDCN) solutions over the initial very short time span so that it has not repeated calculation like in [ – ]
6 Conclusions In this article, we have established the OFDCNI scheme based on the POD technique for the D Sobolev equation
Summary
For convenience and without loss of universality, we think about the following twodimensional ( D) Sobolev equation:. In this article, we use the POD technique to establish an optimized finite difference iterative (OFDCNI) scheme containing very few unknowns but holding sufficiently high accuracy for the D Sobolev equation, analyze the stability and convergence of the OFDCNI solutions, and verify the feasibility and effectiveness of the OFDCNI scheme by means of numerical simulations. The OFDCNI scheme has fully second-order accuracy, is unconditionally stable and absolutely convergent, and is only built by the POD basis constituted with the classical FDCN solutions over the initial very short time span so that it has not repeated calculation like in [ – ] It is development and improvement over the existing ones mentioned above.
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