Elliptic interface boundary value problems play a major role in numerous applications involving heat, fluids, materials, and proteins, to name a few. As an example, in implicit variational solvation, for the construction of biomolecular shapes, the electrostatic contributions satisfy the Poisson-Boltzmann equation with discontinuous dielectric constants across the interface. When interface motions are involved, one often needs not only accurate solution values, but accurate derivatives as well, such as the normal derivatives at the interface. We introduce here the Compact Coupling Interface Method (CCIM), a finite difference method for the elliptic interface problem with interfacial jump conditions. The CCIM can calculate solution values and their derivatives up to second-order accuracy in arbitrary ambient space dimensions. It combines elements of Chern and Shu's Coupling Interface Method and Mayo's approach for elliptic interface boundary value problems, leading to more compact finite difference stencils that are applicable to more general situations. Numerical results on a variety of geometric interfacial shapes and on complex protein molecules in three dimensions support the efficacy of our approach and reveal advantages in accuracy and robustness.
翻译:光滑界面边界值问题在涉及热、流体、材料和蛋白质等多种应用中起着重要作用。 例如,在隐含的变异溶解中,为构建生物分子形状,静电贡献满足了Poisson-Boltzmann等式,在界面中具有不连续的电离电常数。当界面运动涉及到时,人们往往不仅需要准确的解决方案值,而且需要准确的衍生物,例如界面中的普通衍生物。我们在这里引入了《契约组合界面方法》,这是具有间跳跳条件的离子界面问题的有限差异方法。CCIM可以计算解决方案值及其衍生物,最高可达到任意空间维度的二阶精确度。它结合了Chern和Shu的 Coupling接口方法以及Mayo对椭圆界面边界值问题的各种要素,从而导致适用于更一般情况的更紧凑的有限差异。关于几何间形形状和复杂蛋白分子的三个层面的数值结果支持了我们的方法的准确性和准确性。