In this paper, a new alternating direction implicit (ADI) method is introduced to solve potential driven geometric flow PDEs for biomolecular surface generation. For such PDEs, an extra factor is usually added to stabilize the explicit time integration. However, two existing implicit ADI schemes are also based on the scaled form, which involves nonlinear cross derivative terms that have to be evaluated explicitly. This affects the stability and accuracy of these ADI schemes. To overcome these difficulties, we propose a new ADI algorithm based on the unscaled form so that cross derivatives are not involved. Central finite differences are employed to discretize the nonhomogenous diffusion process of the geometric flow. The proposed ADI algorithm is validated through benchmark examples with analytical solutions, reference solutions, or literature results. Moreover, quantitative indicators of a biomolecular surface, including surface area, surface-enclosed volume, and solvation free energy, are analyzed for various proteins. The proposed ADI method is found to be unconditionally stable and more accurate than the existing ADI schemes in all tests. This enables the use of a large time increment in the steady state simulation so that the proposed ADI algorithm is very efficient for biomolecular surface generation.
Recently, a nonlinear Poisson equation has been introduced to model nonlinear and nonlocal hyperpolarization effects in electrostatic solute-solvent interaction for biomolecular solvation analysis. Due to a strong nonlinearity associated with the heterogeneous dielectric media, this Poisson model is difficult to solve numerically, particularly for large protein systems. A new pseudo-transient continuation approach is proposed in this paper to efficiently and stably solve the nonlinear Poisson equation. A Douglas type alternating direction implicit (ADI) method is developed for solving the pseudo-time dependent Poisson equation. Different approximations to the dielectric profile in heterogeneous media are considered in the standard finite difference discretization. The proposed ADI scheme is validated by considering benchmark examples with exact solutions and by solvation analysis of real biomolecules with various sizes. Numerical results are in good agreement with the theoretical prediction, experimental measurements, and those obtained from the boundary value problemapproach. Since the time stability of the proposed ADI scheme can be maintained even using very large time increments, it is efficient for electrostatic analysis involving hyperpolarization effects.
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