arXiv subjects

Adi Ditkowski

Publications and source records attributed to Adi Ditkowski.

14 recordsLinked to original sources

arXiv2024

Error Inhibiting Methods for Finite Elements

Finite Difference methods (FD) are one of the oldest and simplest methods for solving partial differential equations (PDE). Block Finite Difference methods (BFD) are FD methods in which the domain is divided into blocks, or cells, containing two or more grid points, with a different scheme used for each grid point, unlike the standard FD method. It was shown in recent works that BFD schemes might be one to three orders more accurate than their truncation errors. Due to these schemes' ability to inhibit the accumulation of truncation errors, these methods were called Error Inhibiting Schemes (EIS). This manuscript shows that our BFD schemes can be viewed as a particular type of Discontinuous Galerkin (DG) method. Then, we prove the BFD scheme's stability using the standard DG procedure while using a Fourier-like analysis to establish its optimal convergence rate. We present numerical examples in one and two dimensions to demonstrate the efficacy of these schemes.

math.NA↗

arXiv2024

A high order, block finite difference, error inhibiting scheme for the transport equation

We propose a block finite difference, error inhibiting scheme that is fourth-order accurate for short to moderate times and has a six-order convergence rate for long times. This scheme outperforms the standard fourth-order Finite Difference scheme. We also demonstrate that the proposed scheme is a particular type of nodal-based Discontinuous Galerkin method with $p=1$.

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arXiv2020

Error Inhibiting Methods for Finite Elements

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arXiv2020

Error Inhibiting Schemes for Initial Boundary Value Heat Equation

Finite Difference (FD) schemes are widely used in science and engineering for approximating solutions of partial differential equations (PDEs). Error analysis of FD schemes relies on estimating the truncation error at each time step. This approach usually leads to a global error whose order is of the same order of the truncation error. For classical FD schemes the global error is indeed of the same order as the truncation error. A particular class of FD schemes is the Block Finite Difference (BFD) schemes, in which the grid is divided into blocks. The structure of such schemes is similar to the structure of the Discontinuous Galerkin (DG) method, and allows inhabitation of the truncation errors. Recently, much effort was devoted to design BFD schemes whose global error converges faster than the truncation error. In this paper, we elaborate the approach presented in arXiv:1711.07926 for the heat equation with periodic boundary conditions. We generalize this methodology to design BFD schemes for the heat equation with Dirichlet or Neumann boundary conditions, whose global error converges faster than the truncation error. Such schemes are henceforth called Error Inhibiting Schemes. We provide an explicit error analysis, including proofs of stability and convergence of the proposed schemes. We illustrate our approach using several numerical examples, which demonstrate the efficiency of our method in comparison to standard FD schemes.

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arXiv2019

IMEX error inhibiting schemes with post-processing

High order implicit-explicit (IMEX) methods are often desired when evolving the solution of an ordinary differential equation that has a stiff part that is linear and a non-stiff part that is nonlinear. This situation often arises in semi-discretization of partial differential equations and many such IMEX schemes have been considered in the literature. The methods considered usually have a a global error that is of the same order as the local truncation error. More recently, methods with global errors that are one order higher than predicted by the local truncation error have been devised (by Kulikov and Weiner, Ditkowski and Gottlieb). In prior work we investigated the interplay between the local truncation error and the global error to construct explicit and implicit {\em error inhibiting schemes} that control the accumulation of the local truncation error over time, resulting in a global error that is one order higher than expected from the local truncation error, and which can be post-processed to obtain a solution which is two orders higher than expected. In this work we extend our error inhibiting with post-processing framework introduced in our previous work to a class of additive general linear methods with multiple steps and stages. We provide sufficient conditions under which these methods with local truncation error of order p will produce solutions of order (p+1), which can be post-processed to order (p+2), and describe the construction of one such post-processor. We apply this approach to obtain implicit-explicit (IMEX) methods with multiple steps and stages. We present some of our new IMEX methods and show their linear stability properties, and investigate how these methods perform in practice on some numerical test cases.

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arXiv2019

Two-derivative error inhibiting schemes with post-processing

High order methods are often desired for the evolution of ordinary differential equations, in particular those arising from the semi-discretization of partial differential equations. In prior work in we investigated the interplay between the local truncation error and the global error to construct error inhibiting general linear methods (GLMs) that control the accumulation of the local truncation error over time. Furthermore we defined sufficient conditions that allow us to post-process the final solution and obtain a solution that is two orders of accuracy higher than expected from truncation error analysis alone. In this work we extend this theory to the class of two-derivative GLMs. We define sufficient conditions that control the growth of the error so that the solution is one order higher than expected from truncation error analysis, and furthermore define the construction of a simple post-processor that will extract an additional order of accuracy. Using these conditions as constraints, we develop an optimization code that enables us to find explicit two-derivative methods up to eighth order that have favorable stability regions, explicit strong stability preserving methods up to seventh order, and A-stable implicit methods up to fifth order. We numerically verify the order of convergence of a selection of these methods, and the total variation diminishing performance of some of the SSP methods. We confirm that the methods found perform as predicted by the theory developed herein.

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arXiv2019

Explicit and implicit error inhibiting schemes with post-processing

Efficient high order numerical methods for evolving the solution of an ordinary differential equation are widely used. The popular Runge--Kutta methods, linear multi-step methods, and more broadly general linear methods, all have a global error that is completely determined by analysis of the local truncation error. In prior work in we investigated the interplay between the local truncation error and the global error to construct {\em error inhibiting schemes} that control the accumulation of the local truncation error over time, resulting in a global error that is one order higher than expected from the local truncation error. In this work we extend our error inhibiting framework to include a broader class of time-discretization methods that allows an exact computation of the leading error term, which can then be post-processed to obtain a solution that is two orders higher than expected from truncation error analysis. We define sufficient conditions that result in a desired form of the error and describe the construction of the post-processor. A number of new explicit and implicit methods that have this property are given and tested on a variety of ordinary and partial differential equation. We show that these methods provide a solution that is two orders higher than expected from truncation error analysis alone.

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arXiv2019

Loss of Physical Reversibility in Reversible Systems

A dynamical system is said to be reversible if, given an output, the input can always be recovered in a well-posed manner. Nevertheless, we argue that reversible systems that have a time-reversal symmetry, such as the Nonlinear Schr{\"o}dinger equation and the $\phi ^4$ equation can become "physically irreversible". By this, we mean that realistically-small experimental errors in measuring the output can lead to dramatic differences between the recovered input and the original one. The loss of reversibility reveals a natural "arrow of time", reminiscent of the thermodynamic one, which is the direction in which the radiation is emitted outward. Our results are relevant to imaging and reversal applications in nonlinear optics.

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arXiv2018

Loss of Polarization in Collapsing Beams

We show theoretically and demonstrate experimentally that collapsing elliptically-polarized laser beams experience a nonlinear ellipse rotation that is highly sensitive to small fluctuations in the input power. For arbitrarily small fluctuations in the input power and after a sufficiently large propagation distance, the polarization angle becomes uniformly distributed in [0, 2$\pi$] from shot-to-shot. We term this novel phenomenon 'loss of polarization'. We perform experiments in fused-silica glass, nitrogen gas and water, and observe a significant increase in the fluctuations of the output polarization angle for elliptically-polarized femtosecond pulses as the power is increased beyond the critical power for self-focusing. We also show numerically and confirm experimentally that this effect is more prominent in the anomalous group-velocity dispersion (GVD) regime compared to the normal-GVD regime due to the extended lengths of the filaments for the former. Such effects could play an important role in intense-field light-matter interactions in which elliptically-polarized pulses are utilized.

physics.optics↗

arXiv2018

Density Estimation in Uncertainty Propagation Problems Using a Surrogate Model

The effect of uncertainties and noise on a quantity of interest (model output) is often better described by its probability density function (PDF) than by its moments. Although density estimation is a common task, the adequacy of approximation methods (surrogate models) for density estimation has not been analyzed before in the uncertainty-quantification (UQ) literature. In this paper, we first show that standard surrogate models (such as generalized polynomial chaos), which are highly accurate for moment estimation, might completely fail to approximate the PDF, even for one-dimensional noise. This is because density estimation requires that the surrogate model accurately approximates the gradient of the quantity of interest, and not just the quantity of interest itself. Hence, we develop a novel spline-based algorithm for density-estimation whose convergence rate in $L^q$ is polynomial in the sampling resolution. This convergence rate is better than that of standard statistical density-estimation methods (such as histograms and kernel density estimators) at dimensions $1 \leq d\leq \frac{5}{2}m$, where $m$ is the spline order. Furthermore, we obtain the convergence rate for density estimation with any surrogate model that approximates the quantity of interest and its gradient in $L^{\infty}$. Finally, we demonstrate our algorithm for problems in nonlinear optics and fluid dynamics.

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arXiv2017

High Order Finite Difference Schemes for the Heat Equation Whose Convergence Rates are Higher Than Their Truncation Errors

Typically when a semi-discrete approximation to a partial differential equation (PDE) is constructed a discretization of the spatial operator with a truncation error $\tau$ is derived. This discrete operator should be semi-bounded for the scheme to be stable. Under these conditions, the Lax--Ricchtmyer equivalence theorem assures that the scheme converges and that the error will be, at most, of the order of $\| \tau \|$. In most cases, the error is in indeed of the order of $\| \tau \|$. We demonstrate that for the Heat equation stable schemes can be constructed, whose truncation errors are $\tau$, however, the actual errors are much smaller. This gives more degrees of freedom in the design of schemes which can make them more efficient (more accurate or compact) than standard schemes. In some cases, the accuracy of the schemes can be further enhanced using post-processing procedures.

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arXiv2017

Loss of phase and universality of stochastic interactions between laser beams

We show that all laser beams gradually lose their initial phase information in nonlinear propagation. Therefore, if two beams travel a sufficiently long distance before interacting, it is not possible to predict whether they would intersect in- or out-of-phase. Hence, if the underlying propagation model is non-integrable, deterministic predictions and control of the interaction outcome become impossible. Because the relative phase between the two beams becomes uniformly distributed in $[0,2\pi]$, however, the statistics of the interaction outcome are universal, and can be efficiently computed using a polynomial-chaos approach, even when the distributions of the noise sources are unknown.

physics.optics↗

arXiv2017

Error Inhibiting Block One-Step Schemes for Ordinary Differential Equations

The commonly used one step methods and linear multi-step methods all have a global error that is of the same order as the local truncation error (as defined in \cite{gustafsson1995time,quarteroni2010numerical,AllenIsaacson,IsaacsonKeller,Sewell}). In fact, this is true of the entire class of general linear methods. In practice, this means that the order of the method is typically defined solely by the order conditions which are derived by studying the local truncation error. In this work, we investigate the interplay between the local truncation error and the global error, and develop a methodology which defines the construction of explicit {\em error inhibiting} block one-step methods (alternatively written as explicit general linear methods \cite{butcher1993a}). These {\em error inhibiting schemes} are constructed so that the accumulation of the local truncation error over time is controlled, which results in a global error that is one order higher than the local truncation error. In this work, we delineate how to carefully choose the coefficient matrices so that the growth of the local truncation error is inhibited. We then use this theoretical understanding to construct several methods that have higher order global error than local truncation error, and demonstrate their enhanced order of accuracy on test cases. These methods demonstrate that the error inhibiting concept is realizable. Future work will further develop new error inhibiting methods and will analyze the computational efficiency and linear stability properties of these methods.

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arXiv2016

On Spectral Approximations With Nonstandard Weight Functions and Their Implementations to Generalized Chaos Expansions

In this manuscript, we analyze the expansions of functions in orthogonal polynomials associated with a general weight function in a multidimensional setting. Such orthogonal polynomials can be obtained by Gram-Schmidt orthogonalization. However, in most cases, they are not eigenfunctions of some singular Sturm-Liouville problem, as is the case for classical polynomials. Therefore, standard results regarding convergence cannot be applied. Furthermore, since in general, the weight functions are not a tensor product of one-dimensional functions, the orthogonal polynomials are not a tensor product of one-dimensional orthogonal polynomials, as well. In this work, we determine the convergence rate using a comparison Lemma. We also present a spectrally convergent, multidimensional, integration method. Numerical examples demonstrate the efficacy of the proposed method. We show that the use of nonstandard weight functions can allow for efficient integration of singular functions. We also apply this method to Generalized Polynomial Chaos Expansions in the case of dependent random variables.

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