Adaptive Finite Difference Method Research Articles

Uniform convergence analysis of a new adaptive upwind finite difference method for singularly perturbed convection-reaction-diffusion boundary value problems

Uniform convergence analysis of a new adaptive upwind finite difference method for singularly perturbed convection-reaction-diffusion boundary value problems

An adaptive finite-difference method for seismic traveltime modeling based on 3D eikonal equation

3D eikonal equation is a partial differential equation for the calculation of first-arrival traveltimes and has been widely applied in many scopes such as ray tracing, source localization, reflection migration, seismic monitoring and tomographic imaging. In recent years, many advanced methods have been developed to solve the 3D eikonal equation in heterogeneous media. However, there are still challenges for the stable and accurate calculation of first-arrival traveltimes in 3D strongly inhomogeneous media. In this paper, we propose an adaptive finite-difference (AFD) method to numerically solve the 3D eikonal equation. The novel method makes full use of the advantages of different local operators characterizing different seismic wave types to calculate factors and traveltimes, and then the most accurate factor and traveltime are adaptively selected for the convergent updating based on the Fermat principle. Combined with global fast sweeping describing seismic waves propagating along eight directions in 3D media, our novel method can achieve the robust calculation of first-arrival traveltimes with high precision at grid points either near source point or far away from source point even in a velocity model with large and sharp contrasts. Several numerical examples show the good performance of the AFD method, which will be beneficial to many scientific applications.

An Explicit Adaptive Finite Difference Method for the Cahn-Hilliard Equation.

In this study, we propose an explicit adaptive finite difference method (FDM) for the Cahn–Hilliard (CH) equation which describes the process of phase separation. The CH equation has been successfully utilized to model and simulate diverse field applications such as complex interfacial fluid flows and materials science. To numerically solve the CH equation fast and efficiently, we use the FDM and time-adaptive narrow-band domain. For the adaptive grid, we define a narrow-band domain including the interfacial transition layer of the phase field based on an undivided finite difference and solve the numerical scheme on the narrow-band domain. The proposed numerical scheme is based on an alternating direction explicit (ADE) method. To make the scheme conservative, we apply a mass correction algorithm after each temporal iteration step. To demonstrate the superior performance of the proposed adaptive FDM for the CH equation, we present two- and three-dimensional numerical experiments and compare them with those of other previous methods.

A practical adaptive grid method for the Allen–Cahn equation

We present a simple and practical adaptive finite difference method for the Allen–Cahn (AC) equation in the two-dimensional space. We use a temporally adaptive narrow band domain embedded in the uniform discrete rectangular domain. The narrow band domain is defined as a neighboring region of the interface. We employ a recently developed explicit hybrid numerical scheme for the AC equation. Therefore, the computational algorithm on the narrow band discrete domain is simple and fast. We demonstrate the high performance of the proposed adaptive method for the AC equation through various computational experiments.

A fast and practical adaptive finite difference method for the conservative Allen–Cahn model in two-phase flow system

We present a simple and practical adaptive finite difference method for the conservative Allen–Cahn–Navier–Stokes system. For the conservative Allen–Cahn equation, we use a temporally adaptive narrow band domain embedded in the uniform discrete rectangular domain. The narrow band domain is defined as a neighboring region of the interface. The Navier–Stokes equation is solved in a fully discrete domain with the coarse grid than that for the CAC equation. Various benchmark numerical experiments, such as the pressure jump, droplet deformation in shear flow, falling droplet, and rising bubble, are performed to show that the proposed method is efficient and practical for the simulations of two-phase incompressible flow.

An adaptive finite-difference method for accurate simulation of first-arrival traveltimes in heterogeneous media

Accurate simulation of first-arrival traveltimes is the foundation of seismic depth imaging and macro velocity model building for full waveform inversion. In this paper, an adaptive finite-difference method based on Gauss-Seidel iterations has been developed to achieve a more accurate calculation of first-arrival traveltimes in heterogeneous media. The plane wave, spherical wave and refracted wave approximations of local eikonal solver are used to simulate the seismic traveltimes, respectively. Then, a novel updating strategy based on Fermat principle is proposed to make the accurate choice of factor and traveltime, adaptively. The new algorithm can guarantee the high accuracy of traveltime simulation at the grid point not only near the source point, but also at a great distance from the source point. With the combination of global fast sweeping method, the traveltime simulation is unconditionally stable and all possible propagating directions of different seismic waves are considered to describe the complex propagating process due to the velocity model with sharp variations. Several numerical examples were solved to demonstrate that the adaptive finite-difference method can fulfill the accurate, efficient and robust simulation of first-arrival traveltimes.

A VOLATILITY-OF-VOLATILITY EXPANSION OF THE OPTION PRICES IN THE SABR STOCHASTIC VOLATILITY MODEL

We propose a new type of asymptotic expansion for the transition probability density function (or heat kernel) of certain parabolic partial differential equations (PDEs) that appear in option pricing. As other, related methods developed by Costanzino, Hagan, Gatheral, Lesniewski, Pascucci, and their collaborators, among others, our method is based on the computation of the truncated asymptotic expansion of the heat kernel with respect to a “small” parameter. What sets our method apart is that our small parameter is possibly different from the time to expiry and that the resulting commutator calculations go beyond the nilpotent Lie algebra case. In favorable situations, the terms of this asymptotic expansion can quickly be computed explicitly leading to a “closed-form” approximation of the solution, and hence of the option price. Our approximations tend to have much fewer terms than the ones obtained from short time asymptotics, and are thus easier to generalize. Another advantage is that the first term of our expansion corresponds to the classical Black-Scholes model. Our method also provides equally fast approximations of the derivatives of the solution, which is usually a challenge. A full theoretical justification of our method seems very difficult at this time, but we do provide some justification based on the results of (Siyan, Mazzucato, and Nistor, NWEJ 2018). We therefore mostly content ourselves to demonstrate numerically the efficiency of our method by applying it to the solution of the mean-reverting SABR stochastic volatility model PDE, commonly referred to as the [Formula: see text]SABR PDE, by taking the volatility of the volatility parameter [Formula: see text] (vol-of-vol) as a small parameter. For this PDE, we provide extensive numerical tests to gauge the performance of our method. In particular, we compare our approximation to the one obtained using Hagan’s formula and to the one obtained using a new, adaptive finite difference method. We provide an explicit asymptotic expansion for the implied volatility (generalizing Hagan’s formula), which is what is typically needed in concrete applications. We also calibrate our model to observed market option price data. The resulting values for the parameters [Formula: see text], [Formula: see text], and [Formula: see text] are realistic, which provides more evidence for the conjecture that the volatility is mean-reverting.

Numerical Solution of Heston-Hull-White Three-Dimensional PDE with a High Order FD Scheme

A new numerical method for tackling the three-dimensional Heston–Hull–White partial differential equation (PDE) is proposed. This PDE has an application in pricing options when not only the asset price and the volatility but also the risk-free rate of interest are coming from stochastic nature. To solve this time-dependent three-dimensional PDE as efficiently as possible, high order adaptive finite difference (FD) methods are applied for the application of method of lines. It is derived that the new estimates have fourth order of convergence on non-uniform grids. In addition, it is proved that the overall procedure is conditionally time-stable. The results are upheld via several numerical tests.

Fast and accurate adaptive finite difference method for dendritic growth

We propose a fast and accurate adaptive numerical method for solving a phase-field model for dendritic growth. The phase-field model for dendritic growth consists of two equations. One is for capturing the interface between solid and melt. The other is for the temperature distribution. For the phase-field equation, we apply a hybrid explicit method on a time-dependent narrow-band domain, which is defined using the phase-field function. For the temperature equation, we apply the explicit Euler method on the whole computational domain. The novelties of the proposed numerical algorithm are that it is very simple and that it does not require the conventional complex adaptive data structures. Our numerical simulation results are consistent with previous results. Furthermore, the computational time required (CPU time) is shorter.

Pricing multi-asset option problems: a Chebyshev pseudo-spectral method

The aim of this paper is to contribute a new second-order pseudo-spectral method via a non-uniform distribution of the computational nodes for solving multi-asset option pricing problems. In such problems, the prices are required to be as accurately as possible around the strike price. Accordingly, the proposed modification of the Chebyshev–Gauss–Lobatto points would concentrate on this area. This adaptation is also fruitful for the non-smooth payoffs which cause discontinuities in the strike price. The proposed scheme competes well with the existing methods such as finite difference, meshfree, and adaptive finite difference methods on several benchmark problems.

Adaptive implicit finite difference method for natural gas pipeline transient flow

The implicit finite difference method is one of the most widely applied methods for transient natural gas simulation. However, this implicit method is associated with high computational cost. To improve the simulation efficiency of implicit finite difference method, an adaptive strategy is introduced into the simulation process. The proposed adaptive strategy consists of the adaptive time step strategy and the adaptive spatial grid strategy. And these two parts are implemented based on the local error technique and the multilevel grid technique respectively. The results illustrate that the proposed adaptive method can automatically and independently adjust the time step and the spatial grid system according to the gas flow state in the simulation process, and demonstrates a significant advantage in terms of computational accuracy and efficiency compared with the non-adaptive method.

Higher-Order Adaptive Finite Difference Methods for Fully Nonlinear Elliptic Equations

We introduce generalised finite difference methods for solving fully nonlinear elliptic partial differential equations. Methods are based on piecewise Cartesian meshes augmented by additional points along the boundary. This allows for adaptive meshes and complicated geometries, while still ensuring consistency, monotonicity, and convergence. We describe an algorithm for efficiently computing the non-traditional finite difference stencils. We also present a strategy for computing formally higher-order convergent methods. Computational examples demonstrate the efficiency, accuracy, and flexibility of the methods.

A decomposition approach via Fourier sine transform for valuing American knock-out options with rebates

We present an innovative decomposition approach for computing the price and the hedging parameters of American knock-out options with a time-dependent rebate. Our approach is built upon: (i) the Fourier sine transform applied to the partial differential equation with a finite time-dependent spatial domain that governs the option price, and (ii) the decomposition technique that partitions the price of the option into that of the European counterpart and an early exercise premium. Our analytic representations can generalize a number of existing decomposition formulas for some European-style and American-style options. A complexity analysis of the method, together with numerical results, show that the proposed approach is significantly more efficient than the state-of-the-art adaptive finite difference methods, especially in dealing with spot prices near the barrier. Numerical results are also examined in order to provide new insight in the significant effects of the rebate on the option price, the hedging parameters, and the optimal exercise boundary.

Adaptive Finite Difference Methods for Nonlinear Elliptic and Parabolic Partial Differential Equations with Free Boundaries

Monotone finite difference methods provide stable convergent discretizations of a class of degenerate elliptic and parabolic partial differential equations (PDEs). These methods are best suited to regular rectangular grids, which leads to low accuracy near curved boundaries or singularities of solutions. In this article we combine monotone finite difference methods with an adaptive grid refinement technique to produce a PDE discretization and solver which is applied to a broad class of equations, in curved or unbounded domains which include free boundaries. The grid refinement is flexible and adaptive. The discretization is combined with a fast solution method, which incorporates asynchronous time stepping adapted to the spatial scale. The framework is validated on linear problems in curved and unbounded domains. Key applications include the obstacle problem and the one-phase Stefan free boundary problem.

Fast, accurate and robust adaptive finite difference methods for fractional diffusion equations

The computation time required by standard finite difference methods with fixed timesteps for solving fractional diffusion equations is usually very large because the number of operations required to find the solution scales as the square of the number of timesteps. Besides, the solutions of these problems usually involve markedly different time scales, which leads to quite inhomogeneous numerical errors. A natural way to address these difficulties is by resorting to adaptive numerical methods where the size of the timesteps is chosen according to the behaviour of the solution. A key feature of these methods is then the efficiency of the adaptive algorithm employed to dynamically set the size of every timestep. Here we discuss two adaptive methods based on the step-doubling technique. These methods are, in many cases, immensely faster than the corresponding standard method with fixed timesteps and they allow a tolerance level to be set for the numerical errors that turns out to be a good indicator of the actual errors.

Traveling Waves in a One-Dimensional Elastic Continuum Model of Cell Layer Migration with Stretch-Dependent Proliferation

Collective cell migration plays a substantial role in maintaining the cohesion of epithelial cell layers and in wound healing. A number of mathematical models of this process have been developed, all of which reduce to essentially a reaction-diffusion equation with diffusion and proliferation terms that depend on material assumptions about the cell layer. In this paper we extend a one-dimensional mathematical model of cell layer migration of Mi et al. (Biophys. J., 93 (2007), pp. 3745-3752) to incorporate stretch-dependent proliferation, and show that this formulation reduces to a generalized Stefan problem for the density of the layer. We solve numerically the resulting partial differential equation system using an adaptive finite difference method and show that the solutions converge to self-similar or traveling wave solutions. We analyze self-similar solutions for cases with no prolifera- tion, and necessary and sufficient conditions for existence and uniqueness of traveling solutions for a wide range of material assumptions about the cell layer.

Research Spotlights

Accurately estimating solutions for models of physical systems with steep internal layers is an important but difficult numerical problem. For example, the formation of alloys---such as the creation of stainless steel---involves phase separation at the atomic level of its metallic components. This can be successfully modeled, but the delicate sharp interface structure makes computational methods for approximating solutions an ambitious problem. Sharp interface systems arise in the form of elliptic partial differential equations in an array of fluids problems, in semiconductor device simulations, and in microstructure evolution of materials. The Research Spotlights paper in this issue, “Norm-Preserving Discretization of Integral Equations for Elliptic PDEs with Internal Layers I: The One-Dimensional Case,” by Travis Askham and Leslie Greengard, concentrates on the numerical problem of accurate resolution of steep internal layers in one-dimensional problems. Common approaches include adaptive finite difference, finite element, and volume integral methods. This paper concentrates on the third approach, and in particular on adaptive volume integral methods that preserve $L^p$ norm. It considers the effect of the choice of $p$ on conditioning, showing in an example that the conditioning varies significantly with $p$. Well-conditioned discretization methods are shown to require both adaptive refinement within the steep internal layer and a careful choice of the value of $p$. This paper will appeal to SIAM readers with an interest in numerical methods for partial differential equations, materials science, fluid dynamics, or semiconductor physics.

A fully adaptive approximation for quenching‐type reaction‐diffusion equations over circular domains

This article studies a fully adaptive finite difference method for solving quenching‐type nonlinear reaction‐diffusion equations over circular domains. Although an auxiliary condition at the origin and radial symmetry are imposed, adaptations are accomplished via arc‐length‐based monitoring functions in space and time, respectively. The monotonicity and positivity of the numerical solution are proved following a suitable grid constraint, and the numerical stability is ensured in the von Neumann sense. Theoretical bounds of the critical quenching radius are obtained and then refined through the computation. Computational examples are provided to illustrate the effectiveness and plausibility of the new adaptive computational procedure developed. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 472–489, 2014

A hybrid adaptive finite difference method powered by a posteriori error estimation technique

This study focuses on developing an adaptive finite difference method oriented by the a posteriori error estimator that is originally derived from the finite element analysis, building a novel type of hybrid mesh adaptive technique that has not been reported elsewhere as of today. The time-dependent heat conduction problem is chosen for numerical investigation. Corresponding to this target problem, three major numerical “black boxes” are established, namely, finite difference solver, a posteriori error estimation framework, and re-meshing via simultaneous local mesh refinement and re-coarsening. Selected transient heat conduction problems are solved using the present hybrid finite difference mesh adaptive tools. The accuracy of the obtained adaptive solutions appears fairly good while the computational resources can be greatly saved, clearly demonstrating the usefulness, effectiveness, and robustness of the hybrid mesh adaptive approach developed through this study.

Eulerian adaptive finite-difference method for high-velocity impact and penetration problems

Owing to the complex processes involved, faithful prediction of high-velocity impact events demands a simulation method delivering efficient calculations based on comprehensively formulated constitutive models. Such an approach is presented herein, employing a weighted essentially non-oscillatory (WENO) method within an adaptive mesh refinement (AMR) framework for the numerical solution of hyperbolic partial differential equations. Applied widely in computational fluid dynamics, these methods are well suited to the involved locally non-smooth finite deformations, circumventing any requirement for artificial viscosity functions for shock capturing. Application of the methods is facilitated through using a model of solid dynamics based upon hyper-elastic theory comprising kinematic evolution equations for the elastic distortion tensor. The model for finite inelastic deformations is phenomenologically equivalent to Maxwell’s model of tangential stress relaxation. Closure relations tailored to the expected high-pressure states are proposed and calibrated for the materials of interest. Sharp interface resolution is achieved by employing level-set functions to track boundary motion, along with a ghost material method to capture the necessary internal boundary conditions for material interactions and stress-free surfaces. The approach is demonstrated for the simulation of high velocity impacts of steel projectiles on aluminium target plates in two and three dimensions.