arXiv subjects

Sigal Gottlieb

Publications and source records attributed to Sigal Gottlieb.

33 records · Page 2Linked to original sources

arXiv2019

Strong Stability Preserving Integrating Factor Two-step Runge--Kutta Methods

Problems that feature significantly different time scales, where the stiff time-step restriction comes from a linear component, implicit-explicit (IMEX) methods alleviate this restriction if the concern is linear stability. However, where the SSP property is needed, IMEX SSP Runge-Kutta (SSP-IMEX) methods have very restrictive time-steps. An alternative to SSP-IMEX schemes is to adopt an integrating factor approach to handle the linear component exactly and step the transformed problem forward using some time-evolution method. The strong stability properties of integrating factor Runge--Kutta methods were previously established, where it was shown that it is possible to define explicit integrating factor Runge-Kutta methods that preserve strong stability properties satisfied by each of the two components when coupled with forward Euler time-stepping. It was proved that the solution will be SSP if the transformed problem is stepped forward with an explicit SSP Runge-Kutta method that has non-decreasing abscissas. However, explicit SSP Runge-Kutta methods have an order barrier of p=4, and sometimes higher order is desired. In this work we consider explicit SSP two-step Runge--Kutta integrating factor methods to raise the order. We show that strong stability is ensured if the two-step Runge-Kutta method used to evolve the transformed problem is SSP and has non-decreasing abscissas. We find such methods up to eighth order and present their SSP coefficients. Adding a step allows us to break the fourth order barrier on explicit SSP Runge-Kutta methods; furthermore, our explicit SSP two-step Runge--Kutta methods with non-decreasing abscissas typically have larger SSP coefficients than the corresponding one-step methods.

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arXiv2018

A Strong Stability Preserving Analysis for Multistage Two-Derivative Time-Stepping Schemes Based on Taylor Series Conditions

High order strong stability preserving (SSP) time discretizations are often needed to ensure the nonlinear (and sometimes non-inner-product) strong stability properties of spatial discretizations specially designed for the solution of hyperbolic PDEs. Multiderivative time-stepping methods have recently been increasingly used for evolving hyperbolic PDEs, and the strong stability properties of these methods are of interest. In our prior work we explored time discretizations that preserve the strong stability properties of spatial discretizations coupled with forward Euler and a second derivative formulation. However, many spatial discretizations do not satisfy strong stability properties when coupled with this second derivative formulation, but rather with a more natural Taylor series formulation. In this work we demonstrate sufficient conditions for an explicit two-derivative multistage method to preserve the strong stability properties of spatial discretizations in a forward Euler and Taylor series formulation. We call these strong stability preserving Taylor series (SSP-TS) methods. We also prove that the maximal order of SSP-TS methods is p = 6, and define an optimization procedure that allows us to find such SSP methods. Several types of these methods are presented and their efficiency compared. Finally, these methods are tested on several PDEs to demonstrate the benefit of SSP-TS methods, the need for the SSP property, and the sharpness of the SSP time-step in many cases.

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arXiv2018

Downwinding for Preserving Strong Stability in Explicit Integrating Factor Runge--Kutta Methods

Strong stability preserving (SSP) Runge-Kutta methods are desirable when evolving in time problems that have discontinuities or sharp gradients and require nonlinear non-inner-product stability properties to be satisfied. Unlike the case for L2 linear stability, implicit methods do not significantly alleviate the time-step restriction when the SSP property is needed. For this reason, when handling problems with a linear component that is stiff and a nonlinear component that is not, SSP integrating factor Runge--Kutta methods may offer an attractive alternative to traditional time-stepping methods. The strong stability properties of integrating factor Runge--Kutta methods where the transformed problem is evolved with an explicit SSP Runge--Kutta method with non-decreasing abscissas was recently established. In this work, we consider the use of downwinded spatial operators to preserve the strong stability properties of integrating factor Runge--Kutta methods where the Runge--Kutta method has some decreasing abscissas. We present the SSP theory for this approach and present numerical evidence to show that such an approach is feasible and performs as expected. However, we also show that in some cases the integrating factor approach with explicit SSP Runge--Kutta methods with non-decreasing abscissas performs nearly as well, if not better, than with explicit SSP Runge--Kutta methods with downwinding. In conclusion, while the downwinding approach can be rigorously shown to guarantee the SSP property for a larger time-step, in practice using the integrating factor approach by including downwinding as needed with optimal explicit SSP Runge--Kutta methods does not necessarily provide significant benefit over using explicit SSP Runge--Kutta methods with non-decreasing abscissas.

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arXiv2018

Strong Stability Preserving Integrating Factor Runge-Kutta Methods

Strong stability preserving (SSP) Runge-Kutta methods are often desired when evolving in time problems that have two components that have very different time scales. Where the SSP property is needed, it has been shown that implicit and implicit-explicit methods have very restrictive time-steps and are therefore not efficient. For this reason, SSP integrating factor methods may offer an attractive alternative to traditional time-stepping methods for problems with a linear component that is stiff and a nonlinear component that is not. However, the strong stability properties of integrating factor Runge-Kutta methods have not been established. In this work we show that it is possible to define explicit integrating factor Runge-Kutta methods that preserve the desired strong stability properties satisfied by each of the two components when coupled with forward Euler time-stepping, or even given weaker conditions. We define sufficient conditions for an explicit integrating factor Runge--Kutta method to be SSP, namely that they are based on explicit SSP Runge--Kutta methods with non-decreasing abscissas. We find such methods of up to fourth order and up to ten stages, analyze their SSP coefficients, and prove their optimality in a few cases. We test these methods to demonstrate their convergence and to show that the SSP time-step predicted by the theory is generally sharp, and that the non-decreasing abscissa condition is needed in our test cases. Finally, we show that on typical total variation diminishing linear and nonlinear test-cases our new explicit SSP integrating factor Runge-Kutta methods out-perform the corresponding explicit SSP Runge-Kutta methods, implicit-explicit SSP Runge--Kutta methods, and some well-known exponential time differencing methods.

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arXiv2017

Implicit and Implicit-Explicit Strong Stability Preserving Runge-Kutta Methods with High Linear Order

When evolving in time the solution of a hyperbolic partial differential equation, it is often desirable to use high order strong stability preserving (SSP) time discretizations. These time discretizations preserve the monotonicity properties satisfied by the spatial discretization when coupled with the first order forward Euler, under a certain time-step restriction. While the allowable time-step depends on both the spatial and temporal discretizations, the contribution of the temporal discretization can be isolated by taking the ratio of the allowable time-step of the high order method to the forward Euler time-step. This ratio is called the strong stability coefficient. The search for high order strong stability time-stepping methods with high order and large allowable time-step had been an active area of research. It is known that implicit SSP Runge-Kutta methods exist only up to sixth order. However, if we restrict ourselves to solving only linear autonomous problems, the order conditions simplify and we can find implicit SSP Runge-Kutta methods of any linear order. In the current work we aim to find very high linear order implicit SSP Runge-Kutta methods that are optimal in terms of allowable time-step. Next, we formulate an optimization problem for implicit-explicit (IMEX) SSP Runge-Kutta methods and find implicit methods with large linear stability regions that pair with known explicit SSP Runge-Kutta methods of orders plin=3,4,6 as well as optimized IMEX SSP Runge-Kutta pairs that have high linear order and nonlinear orders p=2,3,4. These methods are then tested on sample problems to verify order of convergence and to demonstrate the sharpness of the SSP coefficient and the typical behavior of these methods on test problems.

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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

Explicit Strong Stability Preserving Multistage Two-Derivative Time-Stepping Schemes

High order strong stability preserving (SSP) time discretizations are advantageous for use with spatial discretizations with nonlinear stability properties for the solution of hyperbolic PDEs. The search for high order strong stability time-stepping methods with large allowable strong stability time-step has been an active area of research over the last two decades. Recently, multiderivative time-stepping methods have been implemented with hyperbolic PDEs. In this work we describe sufficient conditions for a two-derivative multistage method to be SSP, and find some optimal SSP multistage two-derivative methods. While explicit SSP Runge--Kutta methods exist only up to fourth order, we show that this order barrier is broken for explicit multi-stage two-derivative methods by designing a three stage fifth order SSP method. These methods are tested on simple scalar PDEs to verify the order of convergence, and demonstrate the need for the SSP condition and the sharpness of the SSP time-step in many cases.

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arXiv2014

A Reduced Radial Basis Function Method for Partial Differential Equations on irregular domains

We propose and test the first Reduced Radial Basis Function Method (R$^2$BFM) for solving parametric partial differential equations on irregular domains. The two major ingredients are a stable Radial Basis Function (RBF) solver that has an optimized set of centers chosen through a reduced-basis-type greedy algorithm, and a collocation-based model reduction approach that systematically generates a reduced-order approximation whose dimension is orders of magnitude smaller than the total number of RBF centers. The resulting algorithm is efficient and accurate as demonstrated through two- and three-dimensional test problems.

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arXiv2014

Parametric Analytical Preconditioning and its Applications to the Reduced Collocation Methods

In this paper, we extend the recently developed reduced collocation method \cite{ChenGottlieb} to the nonlinear case, and propose two analytical preconditioning strategies. One is parameter independent and easy to implement, the other one has the traditional affinity with respect to the parameters which allows for efficient implementation through an offline-online decomposition. Overall, the preconditioning improves the quality of the error estimation uniformly on the parameter domain, and speeds up the convergence of the reduced solution to the truth approximation.

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arXiv2014

Optimal Explicit Strong Stability Preserving Runge--Kutta Methods with High Linear Order and optimal Nonlinear Order

High order spatial discretizations with monotonicity properties are often desirable for the solution of hyperbolic PDEs. These methods can advantageously be coupled with high order strong stability preserving time discretizations. The search for high order strong stability time-stepping methods with large allowable strong stability coefficient has been an active area of research over the last two decades. This research has shown that explicit SSP Runge--Kutta methods exist only up to fourth order. However, if we restrict ourselves to solving only linear autonomous problems, the order conditions simplify and this order barrier is lifted: explicit SSP Runge--Kutta methods of any linear order exist. These methods reduce to second order when applied to nonlinear problems. In the current work we aim to find explicit SSP Runge--Kutta methods with large allowable time-step, that feature high linear order and simultaneously have the optimal fourth order nonlinear order. These methods have strong stability coefficients that approach those of the linear methods as the number of stages and the linear order is increased. This work shows that when a high linear order method is desired, it may be still be worthwhile to use methods with higher nonlinear order.

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arXiv2014

Strong Stability Preserving Multistep Runge-Kutta Methods

High-order spatial discretizations with strong stability properties (such as monotonicity) are desirable for the solution of hyperbolic PDEs. Methods may be compared in terms of the strong stability preserving (SSP) time-step. We prove an upper bound on the SSP coefficient of explicit multistep Runge--Kutta methods of order two and above. Order conditions and monotonicity conditions for such methods are worked out in terms of the method coefficients. Numerical optimization is used to find optimized explicit methods of up to five steps, eight stages, and tenth order. These methods are tested on the advection and Buckley-Leverett equations, and the results for the observed total variation diminishing and positivity preserving time-step are presented.

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arXiv2014

A Fourier Pseudospectral Method for the "Good" Boussinesq Equation with Second Order Temporal Accuracy

In this paper, we discuss the nonlinear stability and convergence of a fully discrete Fourier pseudospectral method coupled with a specially designed second order time-stepping for the numerical solution of the "good" Boussinesq equation. Our analysis improves the existing results presented in earlier literature in two ways. First, an $l_\infty(0, T^*; H2)$ convergence for the solution and $l_\infty(0, T^*; l_2)$ convergence for the time-derivative of the solution are obtained in this paper, instead of the $l_\infty(0, T^*; l_2)$ convergence for the solution and the $l_\infty(0, T^*; H^{-2})$ convergence for the time-derivative, given in [17]. In addition, the stability and convergence of this method is shown to be unconditional for the time step in terms of the spatial grid size, compared with a severe restriction time step restriction $Δt \leq Ch^2$ reported in [17].

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arXiv2012

Reduced Collocation Methods: Reduced Basis Methods in the Collocation Framework

In this paper, we present the first reduced basis method well-suited for the collocation framework. Two fundamentally different algorithms are presented: the so-called Least Squares Reduced Collocation Method (LSRCM) and Empirical Reduced Collocation Method (ERCM). This work provides a reduced basis strategy to practitioners who {prefer} a collocation, rather than Galerkin, approach. Furthermore, the empirical reduced collocation method eliminates a potentially costly online procedure that is needed for non-affine problems with Galerkin approach. Numerical results demonstrate the high efficiency and accuracy of the reduced collocation methods, which match or exceed that of the traditional reduced basis method in the Galerkin framework.

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arXiv2011

Strong stability preserving two-step Runge-Kutta methods

We investigate the strong stability preserving (SSP) property of two-step Runge-Kutta (TSRK) methods. We prove that all SSP TSRK methods belong to a particularly simple subclass of TSRK methods, in which stages from the previous step are not used. We derive simple order conditions for this subclass. Whereas explicit SSP Runge-Kutta methods have order at most four, we prove that explicit SSP TSRK methods have order at most eight. We present TSRK methods of up to eighth order that were found by numerical search. These methods have larger SSP coefficients than any known methods of the same order of accuracy, and may be implemented in a form with relatively modest storage requirements. The usefulness of the TSRK methods is demonstrated through numerical examples, including integration of very high order WENO discretizations.

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arXiv2011

Long time stability of a classical efficient scheme for two dimensional Navier-Stokes equations

We prove that a popular classical implicit-explicit scheme for the 2D incompressible Navier--Stokes equations that treats the viscous term implicitly while the nonlinear advection term explicitly is long time stable provided that the time step is sufficiently small in the case with periodic boundary conditions. The long time stability in the $L^2$ and $H^1$ norms further leads to the convergence of the global attractors and invariant measures of the scheme to those of the NSE itself at vanishing time step. Both semi-discrete in time and fully discrete schemes with either Galerkin Fourier spectral or collocation Fourier spectral methods are considered.

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