Browsing by Author "Sarshar, Arash"

Now showing 1 - 7 of 7
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    Alternating directions implicit integration in a general linear method framework
    Sarshar, Arash; Roberts, Steven; Sandu, Adrian (Elsevier, 2021-05-15)
    Alternating Directions Implicit (ADI) integration is an operator splitting approach to solve parabolic and elliptic partial differential equations in multiple dimensions based on solving sequentially a set of related one-dimensional equations. Classical ADI methods have order at most two, due to the splitting errors. Moreover, when the time discretization of stiff one-dimensional problems is based on Runge–Kutta schemes, additional order reduction may occur. This work proposes a new ADI approach based on the partitioned General Linear Methods framework. This approach allows the construction of high order ADI methods. Due to their high stage order, the proposed methods can alleviate the order reduction phenomenon seen with other schemes. Numerical experiments are shown to provide further insight into the accuracy, stability, and applicability of these new methods.
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    A Fast Time-Stepping Strategy for Dynamical Systems Equipped With a Surrogate Model
    Roberts, Steven; Popov, Andrey A.; Sarshar, Arash; Sandu, Adrian (Society for Industrial & Applied Mathematics (SIAM), 2022-01-01)
    Simulation of complex dynamical systems arising in many applications is computationally challenging due to their size and complexity. Model order reduction, machine learning, and other types of surrogate modeling techniques offer cheaper and simpler ways to describe the dynamics of these systems but are inexact and introduce additional approximation errors. In order to overcome the computational difficulties of the full complex models, on one hand, and the limitations of surrogate models, on the other, this work proposes a new accelerated time-stepping strategy that combines information from both. This approach is based on the multirate infinitesimal general-structure additive Runge–Kutta framework. The inexpensive surrogate model is integrated with a small time step to guide the solution trajectory, and the full model is treated with a large time step to occasionally correct for the surrogate model error and ensure convergence. We provide a theoretical error analysis, and several numerical experiments, to show that this approach can be significantly more efficient than using only the full or only the surrogate model for the integration.
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    Linearly Implicit General Linear Methods
    Sarshar, Arash; Roberts, Steven; Sandu, Adrian (2021-12-01)
    Linearly implicit Runge–Kutta methods provide a fitting balance of implicit treat- ment of stiff systems and computational cost. In this paper we extend the class of linearly implicit Runge–Kutta methods to include multi-stage and multi-step methods. We provide the order con- dition theory to achieve high stage order and overall accuracy while admitting arbitrary Jacobians. Several classes of linearly implicit general linear methods (GLMs) are discussed based on existing families such as type 2 and type 4 GLMs, two-step Runge–Kutta methods, parallel IMEX GLMs, and BDF methods. We investigate the stability implications for stiff problems and provide numerical studies for the behavior of our methods compared to linearly implicit Runge–Kutta methods. Our experiments show nominal order of convergence in test cases where Rosenbrock methods suffer from order reduction.
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    A Numerical Investigation of Matrix-Free Implicit Time-Stepping Methods for Large CFD Simulations
    Sarshar, Arash; Tranquilli, Paul; Pickering, Brent P.; McCall, Andrew; Sandu, Adrian; Roy, Christopher J. (2016)
    This paper is concerned with the development and testing of advanced time-stepping methods suited for the integration of time-accurate, real-world applications of computational fluid dynamics (CFD). The performance of several time discretization methods is studied numerically with regards to computational efficiency, order of accuracy, and stability, as well as the ability to treat effectively stiff problems. We consider matrix-free implementations, a popular approach for time-stepping methods applied to large CFD applications due to its adherence to scalable matrix-vector operations and a small memory footprint. We compare explicit methods with matrix-free implementations of implicit, linearly-implicit, as well as Rosenbrock-Krylov methods. We show that Rosenbrock-Krylov methods are competitive with existing techniques excelling for a number of prob- lem types and settings.
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    Physics-informed neural networks for PDE-constrained optimization and control
    Barry-Straume, Jostein; Sarshar, Arash; Popov, Andrey A.; Sandu, Adrian (2022-05-06)
    A fundamental problem of science is designing optimal control policies that manipulate a given environment into producing the desired outcome. Control PhysicsInformed Neural Networks simultaneously solve a given system state, and its respective optimal control, in a one-stage framework that conforms to the physical laws of the system. Prior approaches use a two-stage framework that models and controls a system sequentially, whereas Control PINNs incorporate the required optimality conditions in their architecture and loss function. The success of Control PINNs is demonstrated by solving the following open-loop optimal control problems: (i) an analytical problem (ii) a one-dimensional heat equation, and (iii) a two-dimensional predator-prey problem.
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    Time Stepping Methods for Multiphysics Problems
    Sarshar, Arash (Virginia Tech, 2021-09-09)
    Mathematical modeling of physical processes often leads to systems of differential and algebraic equations involving quantities of interest. A computer model created based on these equations can be numerically integrated to predict future states of the system and its evolution in time. This thesis investigates current methods in numerical time-stepping schemes, identifying a number of important features needed to speed up and increase the accuracy of the solutions. The focus is on developing new methods suitable for large-scale applications with multiple physical processes, potentially with significant differences in their time-scales. Various families of new methods are introduced with special attention to multirating, low computational cost implicitness, high order of convergence, and robustness. For each family, the order condition theory is discussed and a number of examples are derived. The accuracy and stability of the methods are investigated using standard analysis techniques and numerical experiments are performed to verify the abilities of the new methods.
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    A unified formulation of splitting-based implicit time integration schemes
    Gonzalez-Pinto, Severiano; Hernandez-Abreu, Domingo; Perez-Rodriguez, Maria S.; Sarshar, Arash; Roberts, Steven; Sandu, Adrian (Academic Press – Elsevier, 2022-01-01)
    Splitting-based time integration approaches such as fractional step, alternating direction implicit, operator splitting, and locally one dimensional methods partition the system of interest into components, and solve individual components implicitly in a cost-effective way. This work proposes a unified formulation of splitting time integration schemes in the framework of general-structure additive Runge–Kutta (GARK) methods. Specifically, we develop implicit-implicit (IMIM) GARK schemes, provide the order conditions for this class, and explain their application to partitioned systems of ordinary differential equations. We show that classical splitting methods belong to the IMIM GARK family, and therefore can be studied in this unified framework. New IMIM-GARK splitting methods are developed and tested using parabolic systems.