Darae Jeong

An unconditionally stable hybrid numerical method for solving the Allen-Cahn equation

We present an unconditionally stable second-order hybrid numerical method for solving the Allen-Cahn equation representing a model for antiphase domain coarsening in a binary mixture. The proposed method is based on operator splitting techniques. The Allen-Cahn equation was divided into a linear and a nonlinear equation. First, the linear equation was discretized using a Crank-Nicolson scheme and the resulting discrete system of equations was solved by a fast solver such as a multigrid method. The nonlinear equation was then solved analytically due to the availability of a closed-form solution. Various numerical experiments are presented to confirm the accuracy, efficiency, and stability of the proposed method. In particular, we show that the scheme is unconditionally stable and second-order accurate in both time and space.

A conservative numerical method for the Cahn-Hilliard equation in complex domains

We propose an efficient finite difference scheme for solving the Cahn-Hilliard equation with a variable mobility in complex domains. Our method employs a type of unconditionally gradient stable splitting discretization. We also extend the scheme to compute the Cahn-Hilliard equation in arbitrarily shaped domains. We prove the mass conservation property of the proposed discrete scheme for complex domains. The resulting discretized equations are solved using a multigrid method. Numerical simulations are presented to demonstrate that the proposed scheme can deal with complex geometries robustly. Furthermore, the multigrid efficiency is retained even if the embedded domain is present.

A comparison study of ADI and operator splitting methods on option pricing models

In this paper we perform a comparison study of alternating direction implicit (ADI) and operator splitting (OS) methods on multi-dimensional Black-Scholes option pricing models. The ADI method is used extensively in mathematical finance for numerically solving multi-factor option pricing problems. However, numerical results from the ADI scheme show oscillatory solution behaviors with nonsmooth payoffs or discontinuous derivatives at the exercise price with large time steps. In the ADI scheme, there are source terms which include y-derivatives when we solve x-derivative involving equations. Then, due to the nonsmooth payoffs, source terms contain abrupt changes which are not in the range of implicit discrete operators and this leads to difficulty in solving the problem. On the other hand, the OS method does not contain the other variables derivatives in the source terms. We provide computational results showing the performance of the methods for two-asset option pricing problems. The results show that the OS method is very efficient and gives better accuracy and robustness than the ADI method with large time steps.

A conservative numerical method for the Cahn-Hilliard equation with Dirichlet boundary conditions in complex domains

In this paper we present a conservative numerical method for the Cahn-Hilliard equation with Dirichlet boundary conditions in complex domains. The method uses an unconditionally gradient stable nonlinear splitting numerical scheme to remove the high-order time-step stability constraints. The continuous problem has the conservation of mass and we prove the conservative property of the proposed discrete scheme in complex domains. We describe the implementation of the proposed numerical scheme in detail. The resulting system of discrete equations is solved by a nonlinear multigrid method. We demonstrate the accuracy and robustness of the proposed Dirichlet boundary formulation using various numerical experiments. We numerically show the total energy decrease and the unconditionally gradient stability. In particular, the numerical results indicate the potential usefulness of the proposed method for accurately calculating biological membrane dynamics in confined domains.

An accurate and efficient numerical method for black-scholes equations

We present an ecient and accurate finite-dierence method for computing Black-Scholes partial dierential equations with multi- underlying assets. We directly solve Black-Scholes equations without transformations of variables. We provide computational results showing the performance of the method for two underlying asset option pricing problems.

Fast local image inpainting based on the Allen-Cahn model

In this paper, we propose a fast local image inpainting algorithm based on the Allen-Cahn model. The proposed algorithm is applied only on the inpainting domain and has two features. The first feature is that the pixel values in the inpainting domain are obtained by curvature-driven diffusions and utilizing the image information from the outside of the inpainting region. The second feature is that the pixel values outside of the inpainting region are the same as those in the original input image since we do not compute the outside of the inpainting region. Thus the proposed method is computationally efficient. We split the governing equation into one linear equation and one nonlinear equation by using an operator splitting technique. The linear equation is discretized by using a fully implicit scheme and the nonlinear equation is solved analytically. We prove the unconditional stability of the proposed scheme. To demonstrate the robustness and accuracy of the proposed method, various numerical results on real and synthetic images are presented. We present a fast and efficient algorithm for image inpainting.The proposed model is based on the local Allen-Cahn equation.The proposed scheme is unconditionally stable with respect to time step size.

Finite Element Analysis of Schwarz P Surface Pore Geometries for Tissue-Engineered Scaffolds

Tissue engineering scaffolds provide temporary mechanical support for tissue regeneration. To regenerate tissues more efficiently, an ideal structure of scaffolds should have appropriate porosity and pore structure. In this paper, we generate the Schwarz primitive (P) surface with various volume fractions using a phase-field model. The phase-field model enables us to design various surface-to-volume ratio structures with high porosity and mechanical properties. Comparing the Schwarz P surfaces von Mises stress with that of triply periodic cylinders and cubes, we draw conclusions about the mechanical properties of the Schwarz P surface.

Mathematical model and numerical simulation of the cell growth in scaffolds

A scaffold is a three-dimensional matrix that provides a structural base to fill tissue lesion and provides cells with a suitable environment for proliferation and differentiation. Cell-seeded scaffolds can be implanted immediately or be cultured in vitro for a period of time before implantation. To obtain uniform cell growth throughout the entire volume of the scaffolds, an optimal strategy on cell seeding into scaffolds is important. We propose an efficient and accurate numerical scheme for a mathematical model to predict the growth and distribution of cells in scaffolds. The proposed numerical algorithm is a hybrid method which uses both finite difference approximations and analytic closed-form solutions. The effects of each parameter in the mathematical model are numerically investigated. Moreover, we propose an optimization algorithm which finds the best set of model parameters that minimize a discrete l2 error between numerical and experimental data. Using the mathematical model and its efficient and accurate numerical simulations, we could interpret experimental results and identify dominating mechanisms.

Microphase separation patterns in diblock copolymers on curved surfaces using a nonlocal Cahn-Hilliard equation

Abstract.We investigate microphase separation patterns on curved surfaces in three-dimensional space by numerically solving a nonlocal Cahn-Hilliard equation for diblock copolymers. In our model, a curved surface is implicitly represented as the zero level set of a signed distance function. We employ a discrete narrow band grid that neighbors the curved surface. Using the closest point method, we apply a pseudo-Neumann boundary at the boundary of the computational domain. The boundary treatment allows us to replace the Laplace-Beltrami operator by the standard Laplacian operator. In particular, we can apply standard finite difference schemes in order to approximate the nonlocal Cahn-Hilliard equation in the discrete narrow band domain. We employ a type of unconditionally stable scheme, which was introduced by Eyre, and use the Jacobi iterative to solve the resulting implicit discrete system of equations. In addition, we use the minimum number of grid points for the discrete narrow band domain. Therefore, the algorithm is simple and fast. Numerous computational experiments are provided to study microphase separation patterns for diblock copolymers on curved surfaces in three-dimensional space.Graphical abstract

A finite difference method for a conservative AllenCahn equation on non-flat surfaces

We present an efficient numerical scheme for the conservative AllenCahn (CAC) equation on various surfaces embedded in a narrow band domain in the three-dimensional space. We apply a quasi-Neumann boundary condition on the narrow band domain boundary using the closest point method. This boundary treatment allows us to use the standard Cartesian Laplacian operator instead of the LaplaceBeltrami operator. We apply a hybrid operator splitting method for solving the CAC equation. First, we use an explicit Euler method to solve the diffusion term. Second, we solve the nonlinear term by using a closed-form solution. Third, we apply a spacetime-dependent Lagrange multiplier to conserve the total quantity. The overall scheme is explicit in time and does not need iterative steps; therefore, it is fast. A series of numerical experiments demonstrate the accuracy and efficiency of the proposed hybrid scheme.