BDF Scheme Research Articles

An implicit-explicit approach for atmospheric transport-chemistry problems

We investigate numerical algorithms for use in air pollution models. The emphasis is on the time integration aspects in connection with advection, vertical turbulent diffusion and stiff chemistry. The time integration scheme considered is a second-order implicit-explicit BDF scheme which handles advection explicitly and vertical turbulent diffusion and chemistry implicitly. The investigation is divided into three parts. In the first part we propose a Gauss-Seidel technique for the implicit solution of the chemistry and vertical turbulent diffusion. In the second part we discuss stability properties of the implicit-explicit BDF scheme, assuming the third-order upwind biased finite difference discretization of the advection operator. In the third part we apply the implicit-explicit scheme to a 3D test model and discuss vectorization and parallelization aspects.

Computing brine transport in porous media with an adaptive‐grid method

AbstractAn adaptive‐grid finite‐difference method is applied to a model for non‐isothermal, coupled flow and transport of brine in porous media. In the vicinity of rock salt formations the salt concentration in the fluid becomes large, giving rise to disparate scales in the salt concentrations profiles. A typical situation one encounters is that of a sharp freshwater‐saltwater interface that moves in time. In such situations adaptive‐grid methods are more effective than standard fixed‐grid methods, since they refine the space grid locally and, hence, provide for substantial reduction in the number of grid points, memory use and CPU time. The adaptive‐grid method of this paper is a static, local uniform grid refinement method. Its main feature is that it integrates on nested sequences of locally uniformly refined Cartesian space grids, which are automatically adjusted in time to follow rapid spatial transitions. Variable time steps are used to cope with rapid temporal transitions, including a fast march to possible steady‐state solutions. For time stepping, the implicit, second‐order BDF scheme is used. Two specific example problems are numerically illustrated. The main physical properties involved here are advection and dispersion and in case of dominant advection sharp freshwater‐saltwater interfaces arise.

ODAE methods for the numerical solution of Euler–Lagrange equations

In a series of recent papers [5–7] it is shown that many of the numerical methods for solving Euler–Lagrange equations can be viewed as generalized solutions of an overdetermined differential-algebraic equation (ODAE). For a model with linear constraints, it is shown that if the ODAE is solved by a certain iteration technique in conjunction with BDF discretization, then the corresponding numerical solution coincides with the numerical solution obtained by applying the same BDF scheme to a state-space form of the original Euler–Lagrange equation. In addition, it is shown that using this iteration to solve the ODAE is equivalent to numerically solving a DAE which arises from extending the mechanical system by adding derivatives of the constraints and additional Lagrange multipliers to ensure that those constraints are satisfied. In this paper we examine these equivalences to determine to what extent they continue to hold for problems which are more general than the linear model.

On Numerical Differential Algebraic Problems with Application to Semiconductor Device Simulation

This paper considers questions regarding conditioning of, and numerical methods, for certain differential algebraic equations subject to initial and boundary conditions. The approach taken is that of separating “differential” and “algebraic” solution components, at least theoretically. This yields conditioning results for differential algebraic boundary value problems in terms of “pure” differential problems, for which existing theory is well developed. The process is carried out for problems with (global) index 1 or 2. For semi-explicit boundary value problems of index 1 (where solution components are separated) a convergence theorem is given for a special class of collocation methods. For general index 1 problems advantages and disadvantages of certain symmetric difference schemes are discussed. For initial value problems with index 2 the use of BDF schemes is examined, with a summary of conditions for their successful and stable utilization. Finally, the present considerations and analysis are applied to two problems involving differential algebraic equations arising in semiconductor device simulation.

The Exponential Accuracy of Fourier and Chebyshev Differencing Methods

It is shown that when differencing analytic functions using the pseudospectral Fourier or Chebyshev methods, the error committed decays to zero at an exponential rate.