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Fast and high-order accuracy numerical methods for time-dependent nonlocal problems in $$\mathbb{R}^2$$. (English) Zbl 1445.74054
Summary: In this paper, we study the Crank-Nicolson method for temporal dimension and the piecewise quadratic polynomial collocation method for spatial dimensions of time-dependent nonlocal problems. The new theoretical results of such discretization are that the proposed numerical method is unconditionally stable and its global truncation error is of $$\mathcal{O} (\tau^2+h^{4-\gamma})$$ with $$0<\gamma <1$$, where $$\tau$$ and $$h$$ are the discretization sizes in the temporal and spatial dimensions, respectively. Also we develop the conjugate gradient squared method for solving the resulting discretized nonsymmetric and indefinite systems arising from time-dependent nonlocal problems including two-dimensional cases. By using additive and multiplicative Cauchy kernels in nonlocal problems, structured coefficient matrix-vector multiplication can be performed efficiently in the conjugate gradient squared iteration. Numerical examples are given to illustrate our theoretical results and demonstrate that the computational cost of the proposed method is of $$O(M \log M)$$ operations, where $$M$$ is the number of collocation points.
##### MSC:
 74S99 Numerical and other methods in solid mechanics 74S20 Finite difference methods applied to problems in solid mechanics 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
CGS
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