Browsing by Author "Narayanamurthi, Mahesh"

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    Advanced Time Integration Methods with Applications to Simulation, Inverse Problems, and Uncertainty Quantification
    Narayanamurthi, Mahesh (Virginia Tech, 2020-01-29)
    Simulation and optimization of complex physical systems are an integral part of modern science and engineering. The systems of interest in many fields have a multiphysics nature, with complex interactions between physical, chemical and in some cases even biological processes. This dissertation seeks to advance forward and adjoint numerical time integration methodologies for the simulation and optimization of semi-discretized multiphysics partial differential equations (PDEs), and to estimate and control numerical errors via a goal-oriented a posteriori error framework. We extend exponential propagation iterative methods of Runge-Kutta type (EPIRK) by [Tokman, JCP 2011], to build EPIRK-W and EPIRK-K time integration methods that admit approximate Jacobians in the matrix-exponential like operations. EPIRK-W methods extend the W-method theory by [Steihaug and Wofbrandt, Math. Comp. 1979] to preserve their order of accuracy under arbitrary Jacobian approximations. EPIRK-K methods extend the theory of K-methods by [Tranquilli and Sandu, JCP 2014] to EPIRK and use a Krylov-subspace based approximation of Jacobians to gain computational efficiency. New families of partitioned exponential methods for multiphysics problems are developed using the classical order condition theory via particular variants of T-trees and corresponding B-series. The new partitioned methods are found to perform better than traditional unpartitioned exponential methods for some problems in mild-medium stiffness regimes. Subsequently, partitioned stiff exponential Runge-Kutta (PEXPRK) methods -- that extend stiffly accurate exponential Runge-Kutta methods from [Hochbruck and Ostermann, SINUM 2005] to a multiphysics context -- are constructed and analyzed. PEXPRK methods show full convergence under various splittings of a diffusion-reaction system. We address the problem of estimation of numerical errors in a multiphysics discretization by developing a goal-oriented a posteriori error framework. Discrete adjoints of GARK methods are derived from their forward formulation [Sandu and Guenther, SINUM 2015]. Based on these, we build a posteriori estimators for both spatial and temporal discretization errors. We validate the estimators on a number of reaction-diffusion systems and use it to simultaneously refine spatial and temporal grids.
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    Linearly Implicit Multistep Methods for Time Integration
    Glandon, Steven R.; Narayanamurthi, Mahesh; Sandu, Adrian (Society for Industrial & Applied Mathematics (SIAM), 2022-12)
    Time integration methods for solving initial value problems are an important component of many scientific and engineering simulations. Implicit time integrators are desirable for their stability properties, which significantly relax restrictions on timestep size. However, implicit methods require solutions to one or more systems of nonlinear equations at each timestep, which for large simulations can be prohibitively expensive. This paper introduces a new family of linearly implicit multistep methods (Limm), which only requires the solution of one linear system per timestep. Order conditions and stability theory for these methods are presented, as well as design and implementation considerations. Practical methods of order up to five are developed that have similar error coefficients, but improved stability regions, when compared to the widely used BDF methods. Numerical testing of a self-starting variable stepsize and variable order implementation of the new Limm methods shows measurable performance improvement over a similar BDF implementation.
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    Partitioned exponential methods for coupled multiphysics systems
    Narayanamurthi, Mahesh; Sandu, Adrian (Elsevier, 2021-03-01)
    Multiphysics problems involving two or more coupled physical phenomena are ubiquitous in science and engineering. This work develops a new partitioned exponential approach for the time integration of multiphysics problems. After a possible semi-discretization in space, the class of problems under consideration is modeled by a system of ordinary differential equations where the right-hand side is a summation of two component functions, each corresponding to a given set of physical processes. The partitioned-exponential methods proposed herein evolve each component of the system via an exponential integrator, and information between partitions is exchanged via coupling terms. The traditional approach to constructing exponential methods, based on the variation-of-constants formula, is not directly applicable to partitioned systems. Rather, our approach to developing new partitioned-exponential families is based on a general-structure additive formulation of the schemes. Two method formulations are considered, one based on a linear-nonlinear splitting of the right hand component functions, and another based on approximate Jacobians. The paper develops classical (non-stiff) order conditions theory for partitioned exponential schemes based on particular families of T-trees and B-series theory. Several practical methods of third order are constructed that extend the Rosenbrock-type and EPIRK families of exponential integrators. Several implementation optimizations specific to the application of these methods to reaction-diffusion systems are also discussed. Numerical experiments reveal that the new partitioned-exponential methods can perform better than traditional unpartitioned exponential methods on some problems.