Wenyuan Liao

An implicit fourth-order compact finite difference scheme for one-dimensional Burgers' equation

Abstract A fourth-order compact finite difference method is proposed in this paper to solve one-dimensional Burgers’ equation. The newly proposed method is based on the Hopf–Cole transformation, which transforms the original nonlinear Burgers’ equation into a linear heat equation, and transforms the Dirichlet boundary condition into the Robin boundary condition. The linear heat equation is then solved by an implicit fourth-order compact finite difference scheme. A compact fourth-order formula is also developed to approximate the Robin boundary conditions, while the initial condition for the heat equation is approximated using Simpson’s rule to maintain overall fourth-order accuracy. Numerical experiments have been conducted to demonstrate the efficiency and high-order accuracy of this method. The numerical results also show that the method is unconditionally stable, as there is no constraint on time step size.

Inverse Modeling of Aerosol Dynamics Using Adjoints: Theoretical and Numerical Considerations

In this paper we develop the algorithmic tools needed for inverse modeling of aerosol dynamics. Continuous and discrete adjoints of the aerosol dynamic equation are derived, as well as sensitivity coefficients with respect to the coagulation kernel, the growth rate, and the emission and deposition coefficients. Numerical tests performed in the twin experiment framework for a single component model problem show that the initial distributions and the dynamic parameters can be recovered from time series of observations of particle size distributions.

High-order compact scheme for solving nonlinear Black-Scholes equation with transaction cost

In this paper, an unconditionally stable high-order compact finite difference scheme is proposed. The compact scheme is fourth-order accurate in both the temporal and spatial dimensions. The new method computes both the option price and the hedging delta ∂ V/∂ S simultaneously. Two numerical examples are presented to demonstrate the accuracy and efficiency of the proposed scheme.

An efficient high-order algorithm for solving systems of 3-D reaction-diffusion equations

We discuss an efficient higher order finite difference algorithm for solving systems of 3-D reaction-diffusion equations with nonlinear reaction terms. The algorithm is fourth-order accurate in both the temporal and spatial dimensions. It requires only a regular seven-point difference stencil similar to that used in the standard second-order algorithms, such as the Crank-Nicolson algorithm. Numerical examples are presented to demonstrate the efficiency and accuracy of the new algorithm.

Direct numerical method for an inverse problem of a parabolic partial differential equation

A coefficient inverse problem of the one-dimensional parabolic equation is solved by a high-order compact finite difference method in this paper. The problem of recovering a time-dependent coefficient in a parabolic partial differential equation has attracted considerable attention recently. While many theoretical results regarding the existence and uniqueness of the solution are obtained, the development of efficient and accurate numerical methods is still far from satisfactory. In this paper a fourth-order efficient numerical method is proposed to calculate the function u(x,t) and the unknown coefficient a(t) in a parabolic partial differential equation. Several numerical examples are presented to demonstrate the efficiency and accuracy of the numerical method.

Singular Vector Analysis for Atmospheric Chemical Transport Models

Abstract The singular vectors of a chemical transport model are the directions of maximum perturbation growth over a finite time interval. They have proved useful for the estimation of error growth, the initialization of ensemble forecasts, and the optimal placement of adaptive observations. The aim of this paper is to address computational aspects of singular vector analysis for atmospheric chemical transport models. The distinguishing feature of these models is the presence of stiff chemical interactions. A projection approach to preserve the symmetry of the tangent linear–adjoint operator for stiff systems is discussed, and extended to 3D chemical transport simulations. Numerical results are presented for a simulation of atmospheric pollution in East Asia in March 2001. The singular values and the structure of the singular vectors depend on the length of the simulation interval, the meteorological data, the location of the optimization region and the selection of optimization species, the choice of err...

Inverse modeling of aerosol dynamics: Condensational growth

The feasibility of inverse modeling a multicomponent, size-resolved aerosol evolving by condensation/evaporation is investigated. The adjoint method is applied to the multicomponent aerosol dynamic equation in a box model (zero-dimensional) framework. Both continuous and discrete formulations of the model (the forward equation) and the adjoint are considered. A test example is studied in which the initial aerosol size composition distribution and the pure component vapor concentrations (i.e., vapor pressures) are estimated on the basis of measurements of all species, or a subset of the species, and the entire size distribution, or a portion of the size distribution. It is found that the adjoint method can successfully retrieve the initial size distribution and the pure component vapor concentrations even when only a subset of the species or a portion of the size distribution is observed, although this success is shown to depend upon the form of the initial estimates, the nature of the observations, and the length of the assimilation period. The results presented here provide a basis for the inverse modeling of aerosols in three-dimensional atmospheric chemical transport models.

Adjoint-based optimization of PDEs in moving domains

In this investigation we address the problem of adjoint-based optimization of PDE systems in moving domains. As an example we consider the one-dimensional heat equation with prescribed boundary temperatures and heat fluxes. We discuss two methods of deriving an adjoint system necessary to obtain a gradient of a cost functional. In the first approach we derive the adjoint system after mapping the problem to a fixed domain, whereas in the second approach we derive the adjoint directly in the moving domain by employing methods of the noncylindrical calculus. We show that the operations of transforming the system from a variable to a fixed domain and deriving the adjoint do not commute and that, while the gradient information contained in both systems is the same, the second approach results in an adjoint problem with a simpler structure which is therefore easier to implement numerically. This approach is then used to solve a moving boundary optimization problem for our model system.

An efficient fourth-order low dispersive finite difference scheme for a 2-D acoustic wave equation

In this paper, we propose an efficient fourth-order compact finite difference scheme with low numerical dispersion to solve the two-dimensional acoustic wave equation. Combined with the alternating direction implicit (ADI) technique and Pade approximation, the standard second-order finite difference scheme can be improved to fourth-order and solved as a sequence of one-dimensional problems with high computational efficiency. However such compact higher-order methods suffer from high numerical dispersion. To suppress numerical dispersion, the compact and non-compact stages are interlinked to produce a hybrid scheme, in which the compact stage is based on Pade approximation in both y and temporal dimensions while the non-compact stage is based on Pade approximation in y dimension only. Stability analysis shows that the new scheme is conditionally stable and superior to some existing methods in terms of the Courant-Friedrichs-Lewy (CFL) condition. The dispersion analysis shows that the new scheme has lower numerical dispersion in comparison to the existing compact ADI scheme and the higher-order locally one-dimensional (LOD) scheme. Three numerical examples are solved to demonstrate the accuracy and efficiency of the new method.

On the dispersion, stability and accuracy of a compact higher-order finite difference scheme for 3D acoustic wave equation

Abstract In this paper, we propose a compact fourth-order finite difference scheme with low numerical dispersion to solve the 3D acoustic wave equation. Pade approximation has been used to obtain fourth-order accuracy in both temporal and spatial dimensions, while the alternating direction implicit (ADI) technique has been used to reduce the computational cost. Error analysis has been conducted to show the fourth-order accuracy, which has been confirmed by a numerical example. We have also shown that the proposed method is conditionally stable with a Courant–Friedrichs–Lewy (CFL) condition that is comparable to other existing finite difference schemes. Due to the higher-order accuracy, the new method is found effective in suppressing numerical dispersion.