Equivalence between the DPG method and the exponential integrators for linear parabolic problems

Muñoz‐Matute, Judit; Pardo, David; Demkowicz, Leszek

doi:10.1016/j.jcp.2020.110016

Cited by 15 publications

(41 citation statements)

References 35 publications

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“…In this article, we extend our previous work [35] to linear hyperbolic PDEs. First, we consider a single second-order linear Ordinary Differential Equation (ODE).…”

Section: Introductionmentioning

confidence: 65%

“…As we showed in [35], the equation we obtain for the trace variables is called variationof-constants formula and it is the starting point of exponential integrators [30,32]. Different approximations of this formula lead to different methods [31,33,34].…”

Section: Introductionmentioning

confidence: 99%

“…In [23], authors applied the DPG method in space for the heat equation together with the backward Euler method in time. Recently, in [35], we followed a third approach: to apply the DPG method only in the time variable. In this way, we obtained a DPG-based time-marching scheme for linear transient parabolic PDEs that is compatible with standard Finite Element Method (FEM) based on Bubnov-Galerkin for the space variable.…”

Section: Introductionmentioning

confidence: 99%

“…This work is organized as follows: Section 2 states the variational setting for a single linear second order ODE. Section 3 provides an overview of the ideal DPG method with optimal test functions, we proposed in [35]. We generalize the method in Section 4 for a system of ODEs.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A DPG-based time-marching scheme for linear hyperbolic problems

Muñoz‐Matute

Pardo

Demkowicz

2021

Computer Methods in Applied Mechanics and Engineering

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“…In this article, we extend our previous work [35] to linear hyperbolic PDEs. First, we consider a single second-order linear Ordinary Differential Equation (ODE).…”

Section: Introductionmentioning

confidence: 65%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A DPG-based time-marching scheme for linear hyperbolic problems

Muñoz‐Matute

Pardo

Demkowicz

2021

Computer Methods in Applied Mechanics and Engineering

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“…It has been previously applied to transient problems in the context of space-time formulations or together with finite differences in time [2,3,4]. In this work, we follow the approach of applying the DPG method only in the time variable in order to obtain a DPG-based time-marching scheme for linear transient PDEs [7,8].…”

mentioning

confidence: 99%

DPG-Based Time-Marching-Schemes for Linear Parabolic and Hyperbolic PDEs

Muñoz¹,

Pardo²,

Demkowicz³

2021

10th International Conference on Adaptative Modeling and Simulation

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The Discontinuous Petrov-Galerkin (DPG) method with optimal test functions intends to approximate Partial Differential Equations (PDEs). It was introduced by Demkowicz and Gopalakrishnan in 2010 [1]. The main idea of this method is to select optimal test functions that guarantee the discrete stability of non-coercive problems. For that, they employ test functions that realize the supremum in the inf-sup condition. It has been previously applied to transient problems in the context of space-time formulations or together with finite differences in time [2,3,4]. In this work, we follow the approach of applying the DPG method only in the time variable in order to obtain a DPG-based time-marching scheme for linear transient PDEs [7,8].

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Exploiting the Kronecker product structure of φ−functions in exponential integrators

Muñoz‐Matute

Pardo

Calo

2022

Numerical Meth Engineering

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Exponential time integrators are well-established discretization methods for time semilinear systems of ordinary differential equations. These methods use 𝜑−functions, which are matrix functions related to the exponential. This work introduces an algorithm to speed up the computation of the 𝜑−function action over vectors for two-dimensional (2D) matrices expressed as a Kronecker sum.For that, we present an auxiliary exponential-related matrix function that we express using Kronecker products of one-dimensional matrices. We exploit state-of-the-art implementations of 𝜑−functions to compute this auxiliary function's action and then recover the original 𝜑−action by solving a Sylvester equation system. Our approach allows us to save memory and solve exponential integrators of 2D+time problems in a fraction of the time traditional methods need. We analyze the method's performance considering different linear operators and with the nonlinear 2D+time Allen-Cahn equation.

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Equivalence between the DPG method and the exponential integrators for linear parabolic problems

Cited by 15 publications

References 35 publications

A DPG-based time-marching scheme for linear hyperbolic problems

A DPG-based time-marching scheme for linear hyperbolic problems

DPG-Based Time-Marching-Schemes for Linear Parabolic and Hyperbolic PDEs

Exploiting the Kronecker product structure of φ−functions in exponential integrators

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