Operator splitting ADI schemes for pseudo-time coupled nonlinear solvation simulations

Electrostatic Contributions to the Binding Free Energy of the λcI Repressor to DNA

The coulomb condensate of the nonlinear Poisson-Boltzmann equation: A unified theory

In this paper, a novel compact alternating direction implicit (ADI) scheme is proposed for solving the time-fractional subdiffusion equation in two space dimensions. The established scheme is based on the modified L1 method in time and the compact finite difference method in space. The unique solvability, unconditionally stability and convergence of the scheme are proved. The derived compact ADI scheme is coincident with the one for 2D integer order parabolic equation when the , where is the order of the Riemann–Liouville derivative operator. In addition, the novel ADI scheme is used to solve the 2D modified fractional diffusion equation, and the corresponding stability and convergence results are also given. Numerical results are provided to verify the theoretical analysis.

The linkage between magnesium binding and RNA folding

Stability of ADI schemes applied to convection–diffusion equations with mixed derivative terms

A basis-adaptation method based on polynomial chaos expansion is used for the stochastic nonlinear Poisson---Boltzmann equation. The uncertainty in this numerical approach is motivated by the quantification of noise and fluctuations in nanoscale field-effect sensors. The method used here takes advantage of the properties of the nonlinear Poisson---Boltzmann equation and shows an exact and efficient approximation of the real solution. Numerical examples are motivated by the quantification of noise and fluctuations in nanowire field-effect sensors as a concrete example. Basis adaptation is validated by comparison with the full solution, and it is compared to optimized multi-level Monte-Carlo method, and the model equations are validated by comparison with experiments. Finally, various design parameters of the field-effect sensors are investigated in order to maximize the signal-to-noise ratio.

Compact difference scheme for a class of fractional-in-space nonlinear damped wave equations in two space dimensions

In this work, we developed methods to price both fixed and floating strike Volume Weighted Average Price (VWAP) options. VWAP options have a payoff that is dependent on both the stock price and volume of stock traded over the lifetime of the option. We concentrate on the valuation of European style, fixed and floating strike call options. Little additional effort is required to adapt the results of this work to value puts. First, introductory mathematical material is presented, followed by a general overview of options and how they are priced. The VWAP is then formally introduced, and some examples of VWAP options in practice are given. Expectations and a partial differential equation that describes the price of the options are then constructed. Throughout, we assume the stock price evolves as a geometric Brownian motion process, and the volume is a fast mean reverting process. Upper and lower bounds are established for the price of both fixed and floating strike VWAP options. These bounds are given as analytic formulae and are independent of the volume model. The bounds are obtained from a simple put-call type parity. We then interpret these bounds further. The upper bounds represent the minimum cost of hedging all of the risk. A Monte Carlo investigation is then performed; this gives a vivid picture of how the VWAP options behave. It also allows later results to be benchmarked. In addition to vanilla Monte Carlo, some effective controls variates are found. We conclude the Monte Carlo investigation by finding the Greeks by the finite difference, pathwise, and likelihood methods. Next, approximations to the prices of the VWAP options are obtained by matching the first two moments of the VWAP to a lognormal distribution. For the fixed strike, a partial differential equation similar to the Black-Scholes-Merton one is obtained. When the market price of risk is constant, an analytic expression is found. For a non-constant market price of risk, a numerical method must be used. The floating strike case gives a two dimensional partial differential equation that needs to be solved numerically. In addition to pricing call options, we also demonstrate a method for the valuation of a VWAP digital option. The partial differential equation which describes the prices of the VWAP options is then solved using finite differences. We use explicit, Crank-Nicolson, and Alternating Direction Implicit schemes. Solving by finite differences is a challenging problem as there are four state variables as well as time, and an incomplete set of boundary conditions are known. Additionally, we report several similarity reductions which are effective in simplifying the partial differential equation for the floating strike option. In the final approach to pricing, a series solution to the price is assumed and the first two terms are found. The analysis assumes that the volume process is a fast mean reverting process. The first term of the series is described by a partial differential equation, which is very similar to the one that arises when pricing the Asian option. Numerical experiments established that the first term of the expansion approximates the true solution well when the mean reversion rate is high. Chapter 10 concludes this work with an overview of the results presented in this thesis, along with possible directions for further work.

A second order BDF alternating direction implicit difference scheme for the two-dimensional fractional evolution equation

The full-approximation-storage multigrid strategy has been applied to the heat conduction equation by using several iterative schemes to drive the multilevel process. While aimed at discussing the effectiveness of the most-used solvers when coupled with the multigrid method, this paper includes comparisons between single and multiple grid calculations for both explicit and implicit time-dependent schemes, in addition to stationary ones. An innovative implementation of the alternating direction implicit (ADI) methods is presented, where a noticeable efficiency improvement is obtained through the introduction of an additional smoothing step. The analysis is carried oat on a model problem with mixed Neumann and Dirichlet boundary conditions; then an ADI scheme in conjunction with the multilevel technique is used to compute the temperature distribution in a section of a turbine rotor shaft.

A general numerical method is presented to solve the nonlinear Poisson-Boltzmann (NLPB) equation for an arbitrarily-shaped solute. The essence of the method is the separation of the calculation of the solvent reaction potential from that of the potential due to the ion distribution. The solvent reaction potential is calculated by using an efficient boundary element method. The ion-induced potential is then calculated by means of an efficient volume integration using an iterative solution of the NLPB equation coupled to the fixed molecular and solvent electrostatic potential

Linearized ADI schemes for two-dimensional space-fractional nonlinear Ginzburg–Landau equation

In this paper, the finite difference nonlinear Poisson-Boltzmann (NLPB) equation is used to calculate the electrostatic contribution to the B to Z transition of DNA using detailed molecular structures of each DNA form. The electrostatic transition free energy is described as a balance between the change in intramolecular Coulombic interactions and charge-dependent interactions between the DNA and the solvent. As in many prior studies, we find that the larger electrostatic repulsions among the more closely spaced Z-DNA phosphates destabilize this form compared to B-DNA in the absence of solvent. However, as a result of the more compact three-dimensional geometry of Z-DNA, both water and salt are found to strongly stabilize this conformation to the extent that the total electrostatic free energy favors the B to Z transition in aqueous solution. Water acts not only by screening the inter-phosphate repulsions but also by solvating both charged and polar groups on Z-DNA more favorably than B-DNA. In addition, Z-DNA is stabilized by a substantially higher concentration of nearby counterions than B-DNA. The relative stabilization of Z-DNA by salt increases with increasing bulk salt concentration, leading to the high-salt B to Z transition. We find that the salt dependence of the B to Z transition free energy calculated with the NLPB equation agrees reasonably well with experimental results. Since electrostatic interactions are found to favor the Z-form, nonelectrostatic forces must be responsible for the relative stability of B-DNA in solution. An analysis of these forces suggests that the conformational entropy may play an important role.

This paper addresses the numerical solution of a fractional evolution equation with a weakly singular kernel. The temporal discretization is carried out by the Crank–Nicolson (CN) method based on graded time steps, while the spatial discretization is carried out using an alternating direction implicit (ADI) finite difference method. Stability and convergence rate are also discussed. The convergence rate is $ O(k^2+h_x^2+h_y^2) $, where $ k $ is a parameter of the maximal time steps and $ h_x $, $ h_y $ are the uniform grid steps in space. Three numerical examples are employed to demonstrate the errors and convergence behavior in practice.

The alternating group explicit (AGE) scheme, together with the alternating direction implicit (ADI) scheme, is used for the parabolic equation (PE) method to analyze the electromagnetic scattering from electrically large conducting targets. The calculation can be taken plane by plane by using the finite difference scheme along the paraxial direction. Furthermore, each transverse plane is divided into two regions, which are a small region around the target and an outer region combined with the air layer and a perfect matching layer. The ADI scheme is applied to the small region, whereas the AGE scheme is introduced to the outer region. In this manner, only a few matrix equations are needed to be solved for the small region and the unknowns of the outer region can be directly obtained without solving any matrix equations. Therefore, the CPU time can be reduced significantly by the proposed method and it can be easily adapted to the parallel technology. The numerical results verify the accuracy and efficiency of the proposed method compared with the Crank-Nicolson-based PE method and the ADI-based PE method. Moreover, the parallel efficiency is given as the last numerical example.