Browsing by Author "Tranquilli, Paul"

Now showing 1 - 8 of 8
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    Application of approximate matrix factorization to high order linearly implicit Runge-Kutta methods
    Zhang, H.; Sandu, Adrian; Tranquilli, Paul (Elsevier, 2015-10-01)
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    EPIRK-W and EPIRK-K time discretization methods
    Narayanamurthi, M.; Tranquilli, Paul; Sandu, Adrian; Tokman, M. (2017-01-26)
    Exponential integrators are special time discretization methods where the traditional linear system solves used by implicit schemes are replaced with computing the action of matrix exponential-like functions on a vector. A very general formulation of exponential integrators is offered by the Exponential Propagation Iterative methods of Runge-Kutta type (EPIRK) family of schemes. The use of Jacobian approximations is an important strategy to drastically reduce the overall computational costs of implicit schemes while maintaining the quality of their solutions. This paper extends the EPIRK class to allow the use of inexact Jacobians as arguments of the matrix exponential-like functions. Specifically, we develop two new families of methods: EPIRK-W integrators that can accommodate any approximation of the Jacobian, and EPIRK-K integrators that rely on a specific Krylov-subspace projection of the exact Jacobian. Classical order conditions theories are constructed for these families. A practical EPIRK-W method of order three and an EPIRK-K method of order four are developed. Numerical experiments indicate that the methods proposed herein are computationally favorable when compared to existing exponential integrators.
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    Exponential-Krylov methods for ordinary differential equations
    Tranquilli, Paul; Sandu, Adrian (Academic Press – Elsevier, 2014-12-01)
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    LIRK-W: Linearly-implicit Runge-Kutta methods with approximate matrix factorization
    Tranquilli, Paul; Sandu, Adrian; Zhang, H. (2016-11-22)
    This paper develops a new class of linearly implicit time integration schemes called Linearly-Implicit Runge-Kutta-W (LIRK-W) methods. These schemes are based on an implicit-explicit approach which does not require a splitting of the right hand side and allow for arbitrary, time dependent, and stage varying approximations of the linear systems appearing in the method. Several formulations of LIRK-W schemes, each designed for specific approximation types, and their associated order condition theories are presented.
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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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    Rosenbrock-Krylov Methods for Large Systems of Differential Equations
    Tranquilli, Paul; Sandu, Adrian (Siam Publications, 2014-01-01)
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    Solving stochastic chemical kinetics by Metropolis-Hastings sampling
    Mooasvi, A.; Tranquilli, Paul; Sandu, Adrian (Wilmington Scientific Publisher, Llc, 2016-05-01)
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    Subspace adaptivity in Rosenbrock-Krylov methods for the time integration of initial value problems
    Tranquilli, Paul; Glandon, Ross; Sandu, Adrian (Elsevier, 2021-03-15)
    The Rosenbrock–Krylov family of time integration schemes is an extension of Rosenbrock-W methods that employs a specific Krylov based approximation of the linear system solutions arising within each stage of the integrator. This work proposes an extension of Rosenbrock–Krylov methods to address stability questions which arise for methods making use of inexact linear system solution strategies. Two approaches for improving the stability and efficiency of Rosenbrock–Krylov methods are proposed, one through direct control of linear system residuals and the second through a novel extension of the underlying Krylov space to include stage right hand side vectors. Rosenbrock–Krylov methods employing the new approaches show a substantial improvement in computational efficiency relative to prior implementations.