Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, USA.
Int J Numer Method Biomed Eng. 2014 Apr;30(4):490-516. doi: 10.1002/cnm.2613. Epub 2013 Nov 15.
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.
本文提出了一种新的交替方向隐式 (ADI) 方法，用于求解生物分子表面生成的位势驱动几何流偏微分方程。对于此类偏微分方程，通常需要添加一个额外的因子来稳定显式时间积分。然而，现有的两种隐式 ADI 方案也是基于缩放形式的，这涉及到必须显式评估的非线性交叉导数项。这会影响这些 ADI 方案的稳定性和准确性。为了克服这些困难，我们提出了一种新的基于非缩放形式的 ADI 算法，因此不涉及交叉导数。中心有限差分法用于离散几何流的非齐次扩散过程。通过具有解析解、参考解或文献结果的基准示例验证了所提出的 ADI 算法。此外，还分析了各种蛋白质的生物分子表面的定量指标，包括表面积、表面包围体积和溶剂化自由能。在所有测试中，所提出的 ADI 方法被发现是无条件稳定的，并且比现有的 ADI 方案更准确。这使得在稳态模拟中可以使用较大的时间增量，从而使所提出的 ADI 算法非常高效地用于生物分子表面生成。
