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Chebyshev-Galerkin algorithm for computing 3D Solvation
To solve 3D integral equations of molecular liquids, we have developed a numerical scheme based on the Galerkin method. Using a domain decomposition of the interface region, we reduce the problem to calculations of approximating coefficients and the kernel matrix in spherical shell elements (SSE). Applying the linear transformation of coordinates for each SSE we result in calcualtions of the approximating coefficients in cubic volumes. Using the conventional triple Chebyshev series as a basis set, we derive formulas for calculations of the approximating coefficients and evaluate the computational costs of these operations. We have described the general properties of the Chebyshev-Galerkin matrix and derived analytical expressions for recursion calculations of the matrix elements. We have also outlined an iterative method for the solutions of the nonlinear equations obtained for the approximating coefficients, which is based on the direct inversion in the iterative space. Finally, we have estimated the total computational cost of the proposed scheme and compared it with current algorithms for computing 3D solvation problem. It was found the proposed scheme to be by 2-3 orders effective than the current algorithms based on the uniform FFT.