arXiv subjects

Sigal Gottlieb

Publications and source records attributed to Sigal Gottlieb.

At least 19 recordsLinked to original sources

arXiv2026

Performance Evaluation of Stabilized Corrections for Mixed Precision Runge--Kutta Methods

Mixed precision Runge--Kutta methods reduce the cost of the expensive implicit solves in diagonally implicit Runge--Kutta (DIRK) schemes by evaluating them in low precision, while retaining the accuracy of the scheme for larger time steps. The accuracy lost to the low-precision perturbation can be recovered through inexpensive explicit corrections; however, these corrections have an adverse impact on stability. Recently proposed stabilized corrections remedy this by applying a stabilization matrix to the correction step, but their runtime cost has not previously been quantified. In this work, we present a numerical study of the runtime performance of these stabilized corrections. Using spectral semi-discretizations of two nonlinear partial differential equations, the inviscid Burgers' equation and the porous medium equation, we compare uncorrected mixed precision DIRK methods against explicitly corrected and stabilized variants across half, single, double, and quadruple precision pairings, for SDIRK methods of orders two through four. We report convergence, runtime, and speedups, and show that the stabilized corrections improve the accuracy of the mixed precision schemes while preserving substantial runtime savings. All experiments were performed on an Intel Xeon Platinum 8480+ CPU with Julia version 1.11.4.

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arXiv2026

Smooth perturbations of diagonally implicit Runge--Kutta methods

A mixed accuracy framework for Runge--Kutta methods presented in [Grant, JSC 2022] has been shown to speed up the computation in diagonally implicit Runge--Kutta (DIRK) methods by using less expensive low accuracy approaches for the implicit stages. This theory included both smooth and nonsmooth perturbations, and subsequent work focused primarily on the case of nonsmooth perturbations that arise from mixed precision simulations. In this work the focus is on smooth perturbations that arise from using less accurate models or under-resolved iterative solvers to simplify the implicit computations. We develop an accuracy and stability analysis based on the framework in [Grant, JSC 2022] to design methods that strategically replace the original operator by a lower accuracy operator to reduce computational cost while mitigating the effect of the perturbations. In particular, we focus on designing novel methods that are high order for smooth perturbations that satisfy additional local consistency conditions. Finally, we verify the performance of the novel perturbed DIRK methods designed in this work and numerically study the impact of different types of smooth perturbations on the accuracy and stability of the methods.

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arXiv2026

Stable corrections for perturbed diagonally implicit Runge--Kutta methods

A mixed accuracy framework for Runge--Kutta methods presented in Grant [JSC 2022] and applied to diagonally implicit Runge--Kutta (DIRK) methods can significantly speed up the computation by replacing the implicit solver by less expensive low accuracy approaches such as lower precision computation of the implicit solve, under-resolved iterative solvers, or simpler, less accurate models for the implicit stages. Understanding the effect of the perturbation errors introduced by the low accuracy computations enables the design of stable and accurate mixed accuracy DIRK methods where the errors from the low-accuracy computation are damped out by multiplication by \dt at multiple points in the simulation, resulting in a more accurate simulation than if low-accuracy was used for all computation. To improve upon this, explicit corrections were previously proposed and analyzed for accuracy, and their performance was tested in related work. Explicit corrections work well when the time-step is sufficiently small, but may introduce instabilities when the time-step is larger. In this work, the stability of the mixed accuracy approach is carefully studied, and used to design novel stabilized correction approaches.

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arXiv2026

Mixed precision and mixed accuracy explicit two-derivative Runge--Kutta methods

Mixed precision Runge--Kutta methods have been recently developed and used for the time-evolution of partial differential equations. Two-derivative Runge--Kutta schemes may offer enhanced stability and accuracy properties compared to classical one-derivative methods, making them attractive in a wide variety of problems. However, their computational cost can be significant, motivating the use of a mixed-precision paradigm that employs different floating-point precisions for different function evaluations to balance efficiency and accuracy. To ensure that the perturbations introduced by the low precision computations do not destroy the accuracy of the solution, we need to understand how these perturbation errors propagate. We extend the numerical analysis mixed precision framework previously developed for Runge--Kutta methods to characterize the propagation of the perturbation errors arising from mixed precision computations in explicit and implicit two-derivative Runge--Kutta methods. We use this framework for analyzing the order of the perturbation errors, and for designing new methods that are less sensitive to the effect of the low precision computations. Numerical experiments on linear and nonlinear representative PDEs, demonstrate that appropriately designed mixed-precision two-derivative Runge--Kutta methods achieve the predicted accuracy.

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arXiv2025

Superconvergent Discontinuous Galerkin Method for the Scalar Teukolsky Equation on Hyperboloidal Domains: Efficient Waveform and Self-Force Computation

The long-time evolution of extreme mass-ratio inspiral systems requires minimal phase and dispersion errors to accurately compute far-field waveforms, while high accuracy is essential near the smaller black hole (modeled as a Dirac delta distribution) for self-force computations. Spectrally accurate methods, such as nodal discontinuous Galerkin (DG) methods, are well suited for these tasks. Their numerical errors typically decrease as $\propto (Δx)^{N+1}$, where $Δx$ is the subdomain size and $N$ is the polynomial degree of the approximation. However, certain DG schemes exhibit superconvergence, where truncation, phase, and dispersion errors can decrease as fast as $\propto (Δx)^{2N+1}$. Superconvergent numerical solvers are, by construction, extremely efficient and accurate. We theoretically demonstrate that our DG scheme for the scalar Teukolsky equation with a distributional source is superconvergent, and this property is retained when combined with the hyperboloidal layer compactification technique. This ensures that waveforms, total energy and angular-momentum fluxes, and self-force computations benefit from superconvergence. We empirically verify this behavior across a family of hyperboloidal layer compactifications with varying degrees of smoothness. Additionally, we show that self-force quantities for circular orbits, computed at the point particle's location, also exhibit a certain degree of superconvergence. Our results underscore the potential benefits of numerical superconvergence for efficient and accurate gravitational waveform simulations based on DG methods.

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arXiv2024

A review of high order strong stability preserving two-derivative explicit, implicit, and IMEX methods

High order strong stability preserving (SSP) time discretizations ensure the nonlinear non-inner-product strong stability properties of spatial discretizations suited for the stable simulation of hyperbolic PDEs. Over the past decade multiderivative time-stepping have been used for the time-evolution hyperbolic PDEs, so that the strong stability properties of these methods have become increasingly relevant. In this work we review sufficient conditions for a two-derivative multistage method to preserve the strong stability properties of spatial discretizations in a forward Euler and different conditions on the second derivative. In particular we present the SSP theory for explicit and implicit two-derivative Runge--Kutta schemes, and discuss a special condition on the second derivative under which these implicit methods may be unconditionally SSP. This condition is then used in the context of implicit-explicit (IMEX) multi-derivative Runge--Kutta schemes, where the time-step restriction is independent of the stiff term. Finally, we present the SSP theory for implicit-explicit (IMEX) multi-derivative general linear methods, and some novel second and third order methods where the time-step restriction is independent of the stiff term.

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arXiv2024

Toward exponentially-convergent simulations of extreme-mass-ratio inspirals: A time-domain solver for the scalar Teukolsky equation with singular source terms

Gravitational wave signals from extreme mass ratio inspirals are a key target for space-based gravitational wave detectors. These systems are typically modeled as a distributionally-forced Teukolsky equation, where the smaller black hole is treated as a Dirac delta distribution. Time-domain solvers often use regularization approaches that approximate the Dirac distribution that often introduce small length scales and are a source of systematic error, especially near the smaller black hole. We describe a multi-domain discontinuous Galerkin method for solving the distributionally-forced Teukolsky equation that describes scalar fields evolving on a Kerr spacetime. To handle the Dirac delta, we expand the solution in spherical harmonics and recast the sourced Teukolsky equation as a first-order, one-dimensional symmetric hyperbolic system. This allows us to derive the method's numerical flux to correctly account for the Dirac delta. As a result, our method achieves global spectral accuracy even at the source's location. To connect the near field to future null infinity, we use the hyperboloidal layer method, allowing us to supply outer boundary conditions and providing direct access to the far-field waveform. We document several numerical experiments where we test our method, including convergence tests against exact solutions, energy luminosities for circular orbits, the scheme's superconvergence properties at future null infinity, and the late-time tail behavior of the scalar field. We also compare two systems that arise from different choices of the first-order reduction variables, finding that certain choices are numerically problematic in practice. The methods developed here may be beneficial when computing gravitational self-force effects, where the regularization procedure has been developed for the spherical harmonic modes and high accuracy is needed at the Dirac delta's location.

gr-qc↗

arXiv2023

Discontinuous Galerkin method for linear wave equations involving derivatives of the Dirac delta distribution

Linear wave equations sourced by a Dirac delta distribution $δ(x)$ and its derivative(s) can serve as a model for many different phenomena. We describe a discontinuous Galerkin (DG) method to numerically solve such equations with source terms proportional to $\partial^n δ/\partial x^n$. Despite the presence of singular source terms, which imply discontinuous or potentially singular solutions, our DG method achieves global spectral accuracy even at the source's location. Our DG method is developed for the wave equation written in fully first-order form. The first-order reduction is carried out using a distributional auxiliary variable that removes some of the source term's singular behavior. While this is helpful numerically, it gives rise to a distributional constraint. We show that a time-independent spurious solution can develop if the initial constraint violation is proportional to $δ(x)$. Numerical experiments verify this behavior and our scheme's convergence properties by comparing against exact solutions.

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arXiv2022

Stability Analysis and Performance Evaluation of Mixed-Precision Runge-Kutta Methods

Additive Runge-Kutta methods designed for preserving highly accurate solutions in mixed-precision computation were proposed and analyzed in 4. These specially designed methods use reduced precision for the implicit computations and full precision for the explicit computations. In this work we analyze the stability properties of these methods and their sensitivity to the low precision rounding errors, and demonstrate their performance in terms of accuracy and efficiency. We develop codes in FORTRAN and Julia to solve nonlinear systems of ODEs and PDEs using the mixed precision additive Runge-Kutta (MP-ARK) methods. The convergence, accuracy, runtime, and energy consumption of these methods is explored. We show that for a given level of accuracy, suitably chosen MP-ARK methods may provide significant reductions in runtime.

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arXiv2021

High order strong stability preserving multi-derivative implicit and IMEX Runge--Kutta methods with asymptotic preserving properties

In this work we present a class of high order unconditionally strong stability preserving (SSP) implicit multi-derivative Runge--Kutta schemes, and SSP implicit-explicit (IMEX) multi-derivative Runge--Kutta schemes where the time-step restriction is independent of the stiff term. The unconditional SSP property for a method of order $p>2$ is unique among SSP methods, and depends on a backward-in-time assumption on the derivative of the operator. We show that this backward derivative condition is satisfied in many relevant cases where SSP IMEX schemes are desired. We devise unconditionally SSP implicit Runge--Kutta schemes of order up to $p=4$, and IMEX Runge--Kutta schemes of order up to $p=3$. For the multi-derivative IMEX schemes, we also derive and present the order conditions, which have not appeared previously. The unconditional SSP condition ensures that these methods are positivity preserving, and we present sufficient conditions under which such methods are also asymptotic preserving when applied to a range of problems, including a hyperbolic relaxation system, the Broadwell model, and the Bhatnagar-Gross-Krook (BGK) kinetic equation. We present numerical results to support the theoretical results, on a variety of problems.

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arXiv2021

Performance Evaluation of Mixed-Precision Runge-Kutta Methods

Additive Runge-Kutta methods designed for preserving highly accurate solutions in mixed-precision computation were proposed and analyzed in [8]. These specially designed methods use reduced precision or the implicit computations and full precision for the explicit computations. We develop a FORTRAN code to solve a nonlinear system of ordinary differential equations using the mixed precision additive Runge-Kutta (MP-ARK) methods on IBM POWER9 and Intel x86\_64 chips. The convergence, accuracy, runtime, and energy consumption of these methods is explored. We show that these MP-ARK methods efficiently produce accurate solutions with significant reductions in runtime (and by extension energy consumption).

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arXiv2021

An EIM-degradation free reduced basis method via over collocation and residual hyper reduction-based error estimation

The need for multiple interactive, real-time simulations using different parameter values has driven the design of fast numerical algorithms with certifiable accuracies. The reduced basis method (RBM) presents itself as such an option. RBM features a mathematically rigorous error estimator which drives the construction of a low-dimensional subspace. A surrogate solution is then sought in this low-dimensional space approximating the parameter-induced high fidelity solution manifold. However when the system is nonlinear or its parameter dependence nonaffine, this efficiency gain degrades tremendously, an inherent drawback of the application of the empirical interpolation method (EIM). In this paper, we augment and extend the EIM approach as a direct solver, as opposed to an assistant, for solving nonlinear partial differential equations on the reduced level. The resulting method, called Reduced Over-Collocation method (ROC), is stable and capable of avoiding the efficiency degradation. Two critical ingredients of the scheme are collocation at about twice as many locations as the number of basis elements for the reduced approximation space, and an efficient error indicator for the strategic building of the reduced solution space. The latter, the main contribution of this paper, results from an adaptive hyper reduction of the residuals for the reduced solution. Together, these two ingredients render the proposed R2-ROC scheme both offline- and online-efficient. A distinctive feature is that the efficiency degradation appearing in traditional RBM approaches that utilize EIM for nonlinear and nonaffine problems is circumvented, both in the offline and online stages. Numerical tests on different families of time-dependent and steady-state nonlinear problems demonstrate the high efficiency and accuracy of our R2-ROC and its superior stability performance.

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arXiv2020

A general linear method approach to the design and optimization of efficient, accurate, and easily implemented time-stepping methods in CFD

In simulations of fluid motion time accuracy has proven to be elusive. We seek highly accurate methods with strong enough stability properties to deal with the richness of scales of many flows. These methods must also be easy to implement within current complex, possibly legacy codes. Herein we develop, analyze and test new time stepping methods addressing these two issues with the goal of accelerating the development of time accurate methods addressing the needs of applications. The new methods are created by introducing inexpensive pre-filtering and post-filtering steps to popular methods which have been implemented and tested within existing codes. We show that pre-filtering and post-filtering a multistep or multi-stage method results in new methods which have both multiple steps and stages: these are general linear methods (GLMs). We utilize the well studied properties of GLMs to understand the accuracy and stability of filtered method, and to design optimal new filters for popular time-stepping methods. We present several new embedded families of high accuracy methods with low cognitive complexity and excellent stability properties. Numerical tests of the methods are presented, including ones finding failure points of some methods. Among the new methods presented is a novel pair of alternating filters for the Implicit Euler method which induces a third order, A-stable, error inhibiting scheme which is shown to be particularly effective.

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arXiv2020

A GPU-accelerated mixed-precision WENO method for extremal black hole and gravitational wave physics computations

We develop and use a novel mixed-precision weighted essentially non-oscillatory (WENO) method for solving the Teukolsky equation, which arises when modeling perturbations of Kerr black holes. We show that WENO methods outperform higher-order finite-difference methods, standard in the discretization of the Teukolsky equation, due to the need to add dissipation for stability purposes in the latter. In particular, as the WENO scheme uses no additional dissipation it is well-suited for scenarios requiring long-time evolution such as the study of Price tails and gravitational wave emission from extreme mass ratio binaries. In the mixed-precision approach, the expensive computation of the WENO weights is performed in reduced floating-point precision that results in a significant speedup factor of 3.3. In addition, we use state-of-the-art Nvidia general-purpose graphics processing units and cluster parallelism to further accelerate the WENO computations. Our optimized WENO solver can be used to quickly generate accurate results of significance in the field of black hole and gravitational wave physics. We apply our solver to study the behavior of the Aretakis charge -- a conserved quantity, that if detected by a gravitational wave observatory like LIGO/Virgo would prove the existence of extremal black holes.

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arXiv2020

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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arXiv2020

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

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

L1-ROC and R2-ROC: L1- and R2-based Reduced Over-Collocation methods for parametrized nonlinear partial differential equations

The onerous task of repeatedly resolving certain parametrized partial differential equations (pPDEs) in, e.g. the optimization context, makes it imperative to design vastly more efficient numerical solvers without sacrificing any accuracy. The reduced basis method (RBM) presents itself as such an option. With a mathematically rigorous error estimator, RBM seeks a surrogate solution in a carefully-built subspace of the parameter-induced high fidelity solution manifold. It can improve efficiency by several orders of magnitudes leveraging an offline-online decomposition procedure. However, this decomposition, usually through the empirical interpolation method (EIM) when the PDE is nonlinear or its parameter dependence nonaffine, is either challenging to implement, or severely degrading to the online efficiency. In this paper, we augment and extend the EIM approach in the context of solving pPDEs in two different ways, resulting in the Reduced Over-Collocation methods (ROC). These are stable and capable of avoiding the efficiency degradation inherent to a direct application of EIM. There are two ingredients of these methods. First is a strategy to collocate at about twice as many locations as the number of bases for the surrogate space. The second is an efficient approach for the strategic selection of the parameter values to build the reduced solution space for which we study two choices, a recent empirical L1 approach and a new indicator based on the reduced residual. Together, these two ingredients render the schemes, L1-ROC and R2-ROC, online efficient and immune from the efficiency degradation of EIM for nonlinear and nonaffine problems offline and online. Numerical tests on three different families of nonlinear problems demonstrate the high efficiency and accuracy of these new algorithms and their superior stability performance.

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