The Experts below are selected from a list of 291 Experts worldwide ranked by ideXlab platform
Jeanfrancois Lemieux - One of the best experts on this subject based on the ideXlab platform.
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implementation of newton s method with an analytical jacobian to solve the 1d sea ice Momentum Equation
Journal of Computational Physics, 2017Co-Authors: Jeanpierre Auclair, Jeanfrancois Lemieux, L B Tremblay, H RitchieAbstract:New numerical solvers are being considered in response to the rising computational cost of properly solving the sea ice Momentum Equation at high resolution. The Jacobian free version of Newton's method has allowed models to obtain the converged solution faster than other implicit solvers used previously. To further improve on this recent development, the analytical Jacobian of the 1D sea ice Momentum Equation is derived and used inside Newton's method. The results are promising in terms of computational efficiency. Although robustness remains an issue for some test cases, it is improved compared to the Jacobian free approach. In order to make use of the strong points of both the new and Jacobian free methods, a hybrid preconditioner using the Picard and Jacobian matrices to improve global and local convergence, respectively, is also introduced. This preconditioner combines the robustness and computational efficiency of the previously used preconditioning matrices when solving the sea ice Momentum Equation.
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a comparison of the jacobian free newton krylov method and the evp model for solving the sea ice Momentum Equation with a viscous plastic formulation a serial algorithm study
Journal of Computational Physics, 2012Co-Authors: Jeanfrancois Lemieux, D A Knoll, Bruno Tremblay, David M Holland, Jeanfrancois Lemieux, Martin LoschAbstract:Numerical convergence properties of a recently developed Jacobian-free Newton-Krylov (JFNK) solver are compared to the ones of the widely used EVP model when solving the sea ice Momentum Equation with a Viscous-Plastic (VP) formulation. To do so, very accurate reference solutions are produced with an independent Picard solver with an advective time step of 10s and a tight nonlinear convergence criterion on 10, 20, 40, and 80-km grids. Approximate solutions with the JFNK and EVP solvers are obtained for advective time steps of 10, 20 and 30min. Because of an artificial elastic term, the EVP model permits an explicit time-stepping scheme with a relatively large subcycling time step. The elastic waves excited during the subcycling are intended to damp out and almost entirely disappear such that the approximate solution should be close to the VP solution. Results show that residual elastic waves cause the EVP approximate solution to have notable differences with the reference solution and that these differences get more important as the grid is refined. Compared to the reference solution, additional shear lines and zones of strong convergence/divergence are seen in the EVP approximate solution. The approximate solution obtained with the JFNK solver is very close to the reference solution for all spatial resolutions tested.
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Using the preconditioned Generalized Minimum RESidual (GMRES) method to solve the sea‐ice Momentum Equation
Journal of Geophysical Research, 2008Co-Authors: Jeanfrancois Lemieux, Bruno Tremblay, Stephen J. Thomas, Jan Sedláček, Lawrence A. MysakAbstract:[1] We introduce the preconditioned generalized minimum residual (GMRES) method, along with an outer loop (OL) iteration to solve the sea-ice Momentum Equation. The preconditioned GMRES method is the linear solver. GMRES together with the OL is used to solve the nonlinear Momentum Equation. The GMRES method has low storage requirements, and it is computationally efficient and parallelizable. It was found that the preconditioned GMRES method is about 16 times faster than a stand-alone successive overrelaxation (SOR) solver and three times faster than a stand-alone line SOR (LSOR). Unlike stand-alone SOR and stand-alone LSOR, the cpu time needed by the preconditioned GMRES method for convergence weakly depends on the relaxation parameter when it is smaller than the optimal value. Results also show that with a 6-hour time step, the free drift velocity field is a better initial guess than the previous time step solution. For GMRES, the symmetry of the system matrix is not a prerequisite. The Coriolis term and the off-diagonal part of the water drag term can then be treated implicitly. The implicit treatment eliminates an instability characterized by a residual oscillation in the total kinetic energy of the ice pack that can be present when these off-diagonal terms are handled explicitly. Treating these terms explicitly prevents one from obtaining a high-accuracy solution of the sea-ice Momentum Equation unless a corrector step is applied. In fact, even after a large number of OL iterations, errors in the drift of the same magnitude as the drift itself can be present when these terms are treated explicitly.
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using the preconditioned generalized minimum residual gmres method to solve the sea ice Momentum Equation
Journal of Geophysical Research, 2008Co-Authors: Jeanfrancois Lemieux, Bruno Tremblay, Stephen J. Thomas, Jan Sedláček, Lawrence A. MysakAbstract:[1] We introduce the preconditioned generalized minimum residual (GMRES) method, along with an outer loop (OL) iteration to solve the sea-ice Momentum Equation. The preconditioned GMRES method is the linear solver. GMRES together with the OL is used to solve the nonlinear Momentum Equation. The GMRES method has low storage requirements, and it is computationally efficient and parallelizable. It was found that the preconditioned GMRES method is about 16 times faster than a stand-alone successive overrelaxation (SOR) solver and three times faster than a stand-alone line SOR (LSOR). Unlike stand-alone SOR and stand-alone LSOR, the cpu time needed by the preconditioned GMRES method for convergence weakly depends on the relaxation parameter when it is smaller than the optimal value. Results also show that with a 6-hour time step, the free drift velocity field is a better initial guess than the previous time step solution. For GMRES, the symmetry of the system matrix is not a prerequisite. The Coriolis term and the off-diagonal part of the water drag term can then be treated implicitly. The implicit treatment eliminates an instability characterized by a residual oscillation in the total kinetic energy of the ice pack that can be present when these off-diagonal terms are handled explicitly. Treating these terms explicitly prevents one from obtaining a high-accuracy solution of the sea-ice Momentum Equation unless a corrector step is applied. In fact, even after a large number of OL iterations, errors in the drift of the same magnitude as the drift itself can be present when these terms are treated explicitly.
Martin Losch - One of the best experts on this subject based on the ideXlab platform.
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the adaptive evp method for solving the sea ice Momentum Equation
Ocean Modelling, 2016Co-Authors: Madlen Kimmritz, Sergey Danilov, Martin LoschAbstract:Abstract Stability and convergence of the modified EVP implementation of the visco-plastic sea ice rheology by Bouillon et al., Ocean Modell., 2013, is analyzed on B- and C-grids. It is shown that the implementation on a B-grid is less restrictive with respect to stability requirements than on a C-grid. On C-grids convergence is sensitive to the discretization of the viscosities. We suggest to adaptively vary the parameters of pseudotime subcycling of the modified EVP scheme in time and space to satisfy local stability constraints. This new approach generally improves the convergence of the modified EVP scheme and hence its numerical efficiency. The performance of the new “adaptive EVP” approach is illustrated in a series of experiments with the sea ice component of the MIT general circulation model (MITgcm) that is formulated on a C-grid.
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a comparison of the jacobian free newton krylov method and the evp model for solving the sea ice Momentum Equation with a viscous plastic formulation a serial algorithm study
Journal of Computational Physics, 2012Co-Authors: Jeanfrancois Lemieux, D A Knoll, Bruno Tremblay, David M Holland, Jeanfrancois Lemieux, Martin LoschAbstract:Numerical convergence properties of a recently developed Jacobian-free Newton-Krylov (JFNK) solver are compared to the ones of the widely used EVP model when solving the sea ice Momentum Equation with a Viscous-Plastic (VP) formulation. To do so, very accurate reference solutions are produced with an independent Picard solver with an advective time step of 10s and a tight nonlinear convergence criterion on 10, 20, 40, and 80-km grids. Approximate solutions with the JFNK and EVP solvers are obtained for advective time steps of 10, 20 and 30min. Because of an artificial elastic term, the EVP model permits an explicit time-stepping scheme with a relatively large subcycling time step. The elastic waves excited during the subcycling are intended to damp out and almost entirely disappear such that the approximate solution should be close to the VP solution. Results show that residual elastic waves cause the EVP approximate solution to have notable differences with the reference solution and that these differences get more important as the grid is refined. Compared to the reference solution, additional shear lines and zones of strong convergence/divergence are seen in the EVP approximate solution. The approximate solution obtained with the JFNK solver is very close to the reference solution for all spatial resolutions tested.
Vijay P. Singh - One of the best experts on this subject based on the ideXlab platform.
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Accuracy of kinematic wave and diffusion wave approximations for space-independent flows on infiltrating surfaces with lateral inflow neglected in the Momentum Equation
Hydrological Processes, 1995Co-Authors: Vijay P. SinghAbstract:Error Equations for the kinematic wave and diffusion wave approximations with lateral inflow neglected in the Momentum Equation are derived under simplified conditions for space-independent flows. These Equations specify error as a function of time in the flow hydrograph. The kinematic wave, diffusion wave and dynamic wave solutions are parameterized through a dimensionless parameter γ which is dependent on the initial conditions. This parameter reflects the effect of initial flow depth, channel-bed slope, lateral inflow, infiltration and channel roughness when the initial condition is non-vanishing; it reflects the effect of bed slope, channel roughness and acceleration due to gravity when the initial condition is vanishing. The error Equations are found to be the Riccati Equation. The structure of the error Equations in the case when the Momentum Equation neglects lateral inflow is different from that when the lateral inflow is included.
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Accuracy of kinematic wave and diffusion wave approximations for space‐independent flows with lateral inflow neglected in the Momentum Equation
Hydrological Processes, 1994Co-Authors: Vijay P. SinghAbstract:Error Equations for the kinematic wave and diffusion wave approximations with lateral inflow neglected in the Momentum Equation are derived under simplified conditions for space-independent flows. These Equations specify error as a function of time in the flow hydrograph. The kinematic wave, diffusion wave and dynamic wave solutions are parameterized through a dimensionless parameter γ which is dependent on the initial conditions. This parameter reflects the effect of initial flow depth, channel-bed slope, lateral inflow and channel roughness when the initial condition is non-vanishing; and it reflects the effect of bed slope, channel roughness and acceleration due to gravity when the initial condition is vanishing. The error Equations are found to be the Riccati Equation. The structure of the error Equations in the case when the Momentum Equation neglects lateral inflow is different from that when the lateral inflow is included.
Lawrence A. Mysak - One of the best experts on this subject based on the ideXlab platform.
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Using the preconditioned Generalized Minimum RESidual (GMRES) method to solve the sea‐ice Momentum Equation
Journal of Geophysical Research, 2008Co-Authors: Jeanfrancois Lemieux, Bruno Tremblay, Stephen J. Thomas, Jan Sedláček, Lawrence A. MysakAbstract:[1] We introduce the preconditioned generalized minimum residual (GMRES) method, along with an outer loop (OL) iteration to solve the sea-ice Momentum Equation. The preconditioned GMRES method is the linear solver. GMRES together with the OL is used to solve the nonlinear Momentum Equation. The GMRES method has low storage requirements, and it is computationally efficient and parallelizable. It was found that the preconditioned GMRES method is about 16 times faster than a stand-alone successive overrelaxation (SOR) solver and three times faster than a stand-alone line SOR (LSOR). Unlike stand-alone SOR and stand-alone LSOR, the cpu time needed by the preconditioned GMRES method for convergence weakly depends on the relaxation parameter when it is smaller than the optimal value. Results also show that with a 6-hour time step, the free drift velocity field is a better initial guess than the previous time step solution. For GMRES, the symmetry of the system matrix is not a prerequisite. The Coriolis term and the off-diagonal part of the water drag term can then be treated implicitly. The implicit treatment eliminates an instability characterized by a residual oscillation in the total kinetic energy of the ice pack that can be present when these off-diagonal terms are handled explicitly. Treating these terms explicitly prevents one from obtaining a high-accuracy solution of the sea-ice Momentum Equation unless a corrector step is applied. In fact, even after a large number of OL iterations, errors in the drift of the same magnitude as the drift itself can be present when these terms are treated explicitly.
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using the preconditioned generalized minimum residual gmres method to solve the sea ice Momentum Equation
Journal of Geophysical Research, 2008Co-Authors: Jeanfrancois Lemieux, Bruno Tremblay, Stephen J. Thomas, Jan Sedláček, Lawrence A. MysakAbstract:[1] We introduce the preconditioned generalized minimum residual (GMRES) method, along with an outer loop (OL) iteration to solve the sea-ice Momentum Equation. The preconditioned GMRES method is the linear solver. GMRES together with the OL is used to solve the nonlinear Momentum Equation. The GMRES method has low storage requirements, and it is computationally efficient and parallelizable. It was found that the preconditioned GMRES method is about 16 times faster than a stand-alone successive overrelaxation (SOR) solver and three times faster than a stand-alone line SOR (LSOR). Unlike stand-alone SOR and stand-alone LSOR, the cpu time needed by the preconditioned GMRES method for convergence weakly depends on the relaxation parameter when it is smaller than the optimal value. Results also show that with a 6-hour time step, the free drift velocity field is a better initial guess than the previous time step solution. For GMRES, the symmetry of the system matrix is not a prerequisite. The Coriolis term and the off-diagonal part of the water drag term can then be treated implicitly. The implicit treatment eliminates an instability characterized by a residual oscillation in the total kinetic energy of the ice pack that can be present when these off-diagonal terms are handled explicitly. Treating these terms explicitly prevents one from obtaining a high-accuracy solution of the sea-ice Momentum Equation unless a corrector step is applied. In fact, even after a large number of OL iterations, errors in the drift of the same magnitude as the drift itself can be present when these terms are treated explicitly.
Bruno Tremblay - One of the best experts on this subject based on the ideXlab platform.
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a comparison of the jacobian free newton krylov method and the evp model for solving the sea ice Momentum Equation with a viscous plastic formulation a serial algorithm study
Journal of Computational Physics, 2012Co-Authors: Jeanfrancois Lemieux, D A Knoll, Bruno Tremblay, David M Holland, Jeanfrancois Lemieux, Martin LoschAbstract:Numerical convergence properties of a recently developed Jacobian-free Newton-Krylov (JFNK) solver are compared to the ones of the widely used EVP model when solving the sea ice Momentum Equation with a Viscous-Plastic (VP) formulation. To do so, very accurate reference solutions are produced with an independent Picard solver with an advective time step of 10s and a tight nonlinear convergence criterion on 10, 20, 40, and 80-km grids. Approximate solutions with the JFNK and EVP solvers are obtained for advective time steps of 10, 20 and 30min. Because of an artificial elastic term, the EVP model permits an explicit time-stepping scheme with a relatively large subcycling time step. The elastic waves excited during the subcycling are intended to damp out and almost entirely disappear such that the approximate solution should be close to the VP solution. Results show that residual elastic waves cause the EVP approximate solution to have notable differences with the reference solution and that these differences get more important as the grid is refined. Compared to the reference solution, additional shear lines and zones of strong convergence/divergence are seen in the EVP approximate solution. The approximate solution obtained with the JFNK solver is very close to the reference solution for all spatial resolutions tested.
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Using the preconditioned Generalized Minimum RESidual (GMRES) method to solve the sea‐ice Momentum Equation
Journal of Geophysical Research, 2008Co-Authors: Jeanfrancois Lemieux, Bruno Tremblay, Stephen J. Thomas, Jan Sedláček, Lawrence A. MysakAbstract:[1] We introduce the preconditioned generalized minimum residual (GMRES) method, along with an outer loop (OL) iteration to solve the sea-ice Momentum Equation. The preconditioned GMRES method is the linear solver. GMRES together with the OL is used to solve the nonlinear Momentum Equation. The GMRES method has low storage requirements, and it is computationally efficient and parallelizable. It was found that the preconditioned GMRES method is about 16 times faster than a stand-alone successive overrelaxation (SOR) solver and three times faster than a stand-alone line SOR (LSOR). Unlike stand-alone SOR and stand-alone LSOR, the cpu time needed by the preconditioned GMRES method for convergence weakly depends on the relaxation parameter when it is smaller than the optimal value. Results also show that with a 6-hour time step, the free drift velocity field is a better initial guess than the previous time step solution. For GMRES, the symmetry of the system matrix is not a prerequisite. The Coriolis term and the off-diagonal part of the water drag term can then be treated implicitly. The implicit treatment eliminates an instability characterized by a residual oscillation in the total kinetic energy of the ice pack that can be present when these off-diagonal terms are handled explicitly. Treating these terms explicitly prevents one from obtaining a high-accuracy solution of the sea-ice Momentum Equation unless a corrector step is applied. In fact, even after a large number of OL iterations, errors in the drift of the same magnitude as the drift itself can be present when these terms are treated explicitly.
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using the preconditioned generalized minimum residual gmres method to solve the sea ice Momentum Equation
Journal of Geophysical Research, 2008Co-Authors: Jeanfrancois Lemieux, Bruno Tremblay, Stephen J. Thomas, Jan Sedláček, Lawrence A. MysakAbstract:[1] We introduce the preconditioned generalized minimum residual (GMRES) method, along with an outer loop (OL) iteration to solve the sea-ice Momentum Equation. The preconditioned GMRES method is the linear solver. GMRES together with the OL is used to solve the nonlinear Momentum Equation. The GMRES method has low storage requirements, and it is computationally efficient and parallelizable. It was found that the preconditioned GMRES method is about 16 times faster than a stand-alone successive overrelaxation (SOR) solver and three times faster than a stand-alone line SOR (LSOR). Unlike stand-alone SOR and stand-alone LSOR, the cpu time needed by the preconditioned GMRES method for convergence weakly depends on the relaxation parameter when it is smaller than the optimal value. Results also show that with a 6-hour time step, the free drift velocity field is a better initial guess than the previous time step solution. For GMRES, the symmetry of the system matrix is not a prerequisite. The Coriolis term and the off-diagonal part of the water drag term can then be treated implicitly. The implicit treatment eliminates an instability characterized by a residual oscillation in the total kinetic energy of the ice pack that can be present when these off-diagonal terms are handled explicitly. Treating these terms explicitly prevents one from obtaining a high-accuracy solution of the sea-ice Momentum Equation unless a corrector step is applied. In fact, even after a large number of OL iterations, errors in the drift of the same magnitude as the drift itself can be present when these terms are treated explicitly.
