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D A Knoll - One of the best experts on this subject based on the ideXlab platform.
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an efficient physics based preconditioner for the fully Implicit Solution of small scale thermally driven atmospheric flows
Journal of Computational Physics, 2003Co-Authors: Jon M Reisner, Andrzej A Wyszogrodzki, Vincent A Mousseau, D A KnollAbstract:In atmospheric flow situations typical of a small-scale atmospheric thermal, a separation of time scales exists between the fast sound wave time scale and the advective time scale. Atmospheric models have been designed to take advantage of this disparity of time scales with numerical approaches such as the semi-Implicit or split-explicit approach being used to efficiently step over the fast sound waves. Some of these numerical approaches are first order in time. To improve accuracy over these methods, a fully Implicit and nonlinearly consistent (INC) flow solver has been developed for the Navier-Stokes equation set. In our INC method, the equation set is solved by use of the Jacobian-free Newton-Krylov (JFNK) method. An efficient preconditioner has been developed which uses the semi-Implicit method to solve the governing equations. Being that this preconditioner was designed to attack the fastest waves in the system and not other features in the Implicit system such as advection or turbulent diffusion, the preconditioning technique is labeled as a physics-based preconditioner. A variety of linear solvers including SSOR, Krylov methods and/or multigrid approaches are used to approximately invert the pressure matrix in the semi-Implicit algorithm. A suite of simulations will be conducted utilizing different linear solvers for the simple problem of the bouyant rise of a warm bubble. The problem will also document the ability of the INC approach to achieve second order in time accuracy.
Dimitri J. Mavriplis - One of the best experts on this subject based on the ideXlab platform.
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Implicit Solution of the unsteady euler equations for high order accurate discontinuous galerkin discretizations
Journal of Computational Physics, 2007Co-Authors: Li Wang, Dimitri J. MavriplisAbstract:Efficient Solution techniques for high-order accurate time-dependent problems are investigated for solving the two-dimensional non-linear Euler equations in this work. The spatial discretization consists of a high-order accurate discontinuous Galerkin (DG) approach. Implicit time-integration techniques are considered exclusively in order to avoid the stability restrictions of explicit methods. Standard backwards differencing methods (BDF1 and BDF2) as well as a second-order Crank-Nicholson (CN2) and a fourth-order Implicit Runge-Kutta (IRK4) scheme are considered in an attempt to balance the spatial and temporal accuracy of the overall approach. The Implicit system arising at each time step is solved using a p-multigrid approach, which is shown to produce h independent convergence rates, while remaining relatively insensitive to the time-step size. The Crank-Nicholson methodology, although not L-stable, demonstrates superior performance compared to the BDF2 scheme for the problems chosen in this work. However, the fourth-order accurate Implicit Runge-Kutta scheme is found to be the most efficient in terms of computational cost for a given accuracy level as compared to the lower-order schemes, in spite of the added cost per time step, and the benefits of this scheme increase for tighter error tolerances.
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Implicit Solution of the unsteady Euler equations for high-order accurate discontinuous Galerkin discretizations
Journal of Computational Physics, 2007Co-Authors: Li Wang, Dimitri J. MavriplisAbstract:Efficient Solution techniques for high-order accurate time-dependent problems are investigated for solving the two-dimensional non-linear Euler equations in this work. The spatial discretization consists of a high-order accurate discontinuous Galerkin (DG) approach. Implicit time-integration techniques are considered exclusively in order to avoid the stability restrictions of explicit methods. Standard backwards differencing methods (BDF1 and BDF2) as well as a second-order Crank-Nicholson (CN2) and a fourth-order Implicit Runge-Kutta (IRK4) scheme are considered in an attempt to balance the spatial and temporal accuracy of the overall approach. The Implicit system arising at each time step is solved using a p-multigrid approach, which is shown to produce h independent convergence rates, while remaining relatively insensitive to the time-step size. The Crank-Nicholson methodology, although not L-stable, demonstrates superior performance compared to the BDF2 scheme for the problems chosen in this work. However, the fourth-order accurate Implicit Runge-Kutta scheme is found to be the most efficient in terms of computational cost for a given accuracy level as compared to the lower-order schemes, in spite of the added cost per time step, and the benefits of this scheme increase for tighter error tolerances. © 2007 Elsevier Inc. All rights reserved.
Jon M Reisner - One of the best experts on this subject based on the ideXlab platform.
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an efficient physics based preconditioner for the fully Implicit Solution of small scale thermally driven atmospheric flows
Journal of Computational Physics, 2003Co-Authors: Jon M Reisner, Andrzej A Wyszogrodzki, Vincent A Mousseau, D A KnollAbstract:In atmospheric flow situations typical of a small-scale atmospheric thermal, a separation of time scales exists between the fast sound wave time scale and the advective time scale. Atmospheric models have been designed to take advantage of this disparity of time scales with numerical approaches such as the semi-Implicit or split-explicit approach being used to efficiently step over the fast sound waves. Some of these numerical approaches are first order in time. To improve accuracy over these methods, a fully Implicit and nonlinearly consistent (INC) flow solver has been developed for the Navier-Stokes equation set. In our INC method, the equation set is solved by use of the Jacobian-free Newton-Krylov (JFNK) method. An efficient preconditioner has been developed which uses the semi-Implicit method to solve the governing equations. Being that this preconditioner was designed to attack the fastest waves in the system and not other features in the Implicit system such as advection or turbulent diffusion, the preconditioning technique is labeled as a physics-based preconditioner. A variety of linear solvers including SSOR, Krylov methods and/or multigrid approaches are used to approximately invert the pressure matrix in the semi-Implicit algorithm. A suite of simulations will be conducted utilizing different linear solvers for the simple problem of the bouyant rise of a warm bubble. The problem will also document the ability of the INC approach to achieve second order in time accuracy.
Li Wang - One of the best experts on this subject based on the ideXlab platform.
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Implicit Solution of the unsteady euler equations for high order accurate discontinuous galerkin discretizations
Journal of Computational Physics, 2007Co-Authors: Li Wang, Dimitri J. MavriplisAbstract:Efficient Solution techniques for high-order accurate time-dependent problems are investigated for solving the two-dimensional non-linear Euler equations in this work. The spatial discretization consists of a high-order accurate discontinuous Galerkin (DG) approach. Implicit time-integration techniques are considered exclusively in order to avoid the stability restrictions of explicit methods. Standard backwards differencing methods (BDF1 and BDF2) as well as a second-order Crank-Nicholson (CN2) and a fourth-order Implicit Runge-Kutta (IRK4) scheme are considered in an attempt to balance the spatial and temporal accuracy of the overall approach. The Implicit system arising at each time step is solved using a p-multigrid approach, which is shown to produce h independent convergence rates, while remaining relatively insensitive to the time-step size. The Crank-Nicholson methodology, although not L-stable, demonstrates superior performance compared to the BDF2 scheme for the problems chosen in this work. However, the fourth-order accurate Implicit Runge-Kutta scheme is found to be the most efficient in terms of computational cost for a given accuracy level as compared to the lower-order schemes, in spite of the added cost per time step, and the benefits of this scheme increase for tighter error tolerances.
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Implicit Solution of the unsteady Euler equations for high-order accurate discontinuous Galerkin discretizations
Journal of Computational Physics, 2007Co-Authors: Li Wang, Dimitri J. MavriplisAbstract:Efficient Solution techniques for high-order accurate time-dependent problems are investigated for solving the two-dimensional non-linear Euler equations in this work. The spatial discretization consists of a high-order accurate discontinuous Galerkin (DG) approach. Implicit time-integration techniques are considered exclusively in order to avoid the stability restrictions of explicit methods. Standard backwards differencing methods (BDF1 and BDF2) as well as a second-order Crank-Nicholson (CN2) and a fourth-order Implicit Runge-Kutta (IRK4) scheme are considered in an attempt to balance the spatial and temporal accuracy of the overall approach. The Implicit system arising at each time step is solved using a p-multigrid approach, which is shown to produce h independent convergence rates, while remaining relatively insensitive to the time-step size. The Crank-Nicholson methodology, although not L-stable, demonstrates superior performance compared to the BDF2 scheme for the problems chosen in this work. However, the fourth-order accurate Implicit Runge-Kutta scheme is found to be the most efficient in terms of computational cost for a given accuracy level as compared to the lower-order schemes, in spite of the added cost per time step, and the benefits of this scheme increase for tighter error tolerances. © 2007 Elsevier Inc. All rights reserved.
John B Drake - One of the best experts on this subject based on the ideXlab platform.
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accuracy analysis of a spectral element atmospheric model using a fully Implicit Solution framework
Monthly Weather Review, 2010Co-Authors: Katherine J Evans, Mark A Taylor, John B DrakeAbstract:Abstract A fully Implicit (FI) time integration method has been implemented into a spectral finite-element shallow-water equation model on a sphere, and it is compared to existing fully explicit leapfrog and semi-Implicit methods for a suite of test cases. This experiment is designed to determine the time step sizes that minimize simulation time while maintaining sufficient accuracy for these problems. For test cases without an analytical Solution from which to compare, it is demonstrated that time step sizes 30–60 times larger than the gravity wave stability limits and 6–20 times larger than the advective-scale stability limits are possible using the FI method without a loss in accuracy, depending on the problem being solved. For a steady-state test case, the FI method produces error within machine accuracy limits as with existing methods, but using an arbitrarily large time step size.
