A unified approach for deriving optimal finite differences

Abstract

A unified approach to derive optimal finite differences is presented which combines three critical elements for numerical performance especially for multi-scale physical problems, namely, order of accuracy, spectral resolution and stability. The resulting mathematical framework reduces to a minimization problem subjected to equality and inequality constraints. We show that the framework can provide analytical results for optimal schemes and their numerical performance including, for example, the type of errors that appear for spectrally optimal schemes. By coupling the problem in this unified framework, one can effectively decouple the requirements for order of accuracy and spectral resolution, for example. Alternatively, we show how the framework exposes the tradeoffs between e.g. accuracy and stability and how this can be used to construct explicit schemes that remain stable with very large time steps. We also show how spectrally optimal schemes only bias odd-order derivatives to remain stable, at the expense of accuracy, while leaving even-order derivatives with symmetric coefficients. Schemes constructed within this framework are tested for diverse model problems with an emphasis on reproducing the physics accurately.schemes are decoupled and optimized separately in order to achieve maximum resolving efficiency for both the operators. The optimization of the temporal scheme is also subject to the stability constraint. Schemes that remain stable for a broader range in the appropriate parameter space (time step size, grid spacing, nondimensional groups such as CFL number, etc.) are typically preferred as simulations with larger time steps are computationally less expensive. While the time step and grid spacing are conventionally subject to the stability and resolution requirements, [13] shows that optimal values for these that minimize computational cost for some error level can also be obtained.An overriding question is, thus, whether it is possible to find optimal schemes of given order, that are stable and that minimize the spectral error in a suitably defined manner typically informed by the physical characteristics of the problem being solved. This is the main motivation of the present work. The mathematical framework in which this can be achieved reduces to an optimization problem with equality and inequality constraints which can be solved, under certain conditions, analytically. The importance of this work is that it allows us to express physically meangingful desired properties and constraints into a unified mathematical framework which results in highly-accurate schemes for a particular problem of interest. Another important aspect of the proposed framework is that it also exposes explicitly tradeoffs that can be profitably used in specific circumstances. For example, we show that it is possible to construct explicit schemes that can remain stable for very large time steps (even an order of magnitude larger than equivalent standard schemes) when constraints on accuracy at some scales can be relaxed. W...

A unified framework to generate optimized compact finite difference schemes

“…Kumari, et al (2019) proposed a unified approach to derive optimized explicit schemes in [1]. In this paper, we present a unified framework to derive optimized compact schemes as a natural extension and generalization of the approach presented in [1]. We also show that optimized schemes derived using the explicit formulation can be recovered as special cases of the compact formulation presented in this paper.…”

“…Consequently, for this case we recover the non-optimized standard scheme. In other words, O 1 1 (4) is identical to the non-optimized S 1 1 (4). The first row of Fig.…”

“…Thus, depending upon the problem at hand, one can choose to use explicit or compact formulation. Kumari, et al (2019) proposed a unified approach to derive optimized explicit schemes in [1]. In this paper, we present a unified framework to derive optimized compact schemes as a natural extension and generalization of the approach presented in [1].…”

“…The method of matching coefficients does not guarantee the desired spectral behavior of the compact scheme derived from Eq. (1). To ensure that all the relevant scales in a problem are properly resolved, it is crucial to have desired spectral accuracy at the wavenumbers of interest.…”

“…Frameworks are requirement specific and there is no unifying generalized framework. For example, [3][4][5][6][7] focus on pentadiagonal compact schemes, [1,[9][10][11][12] are exclusively for explicit finite difference, and [3,[9][10][11] approximate only the first derivative.…”