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- Author or Editor: R. de Villiers x
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Abstract
An augmented Lagrangian multiplier-penalty method is applied for the first time to solving the problem of enforcing simultaneous conservation of the nonlinear integral invariants of the shallow water equations on a limited-area domain. The method approximates the nonlinearly constrained minimization problem by solving a series of unconstrained minimization problems.
The computational efficiency and accuracy of the method is tested using two finite-difference solvers of the nonlinear shallow water equations on a β-plane. The method is also compared with a pure quadratic penalty approach. The updating of the Lagrangian multipliers and the penalty parameters is done using procedures suggested by Bertsekas. The method yielded satisfactory results in the conservation of the integral constraints while the additional CPU time required did not exceed 15% of the total CPU time spent on the numerical solution of the shallow water equations. The methods proved to be simple in their implementation and they have a broad scope of applicability to other problems involving nonlinear constraints; for instance, the variational nonlinear normal mode initialization.
Abstract
An augmented Lagrangian multiplier-penalty method is applied for the first time to solving the problem of enforcing simultaneous conservation of the nonlinear integral invariants of the shallow water equations on a limited-area domain. The method approximates the nonlinearly constrained minimization problem by solving a series of unconstrained minimization problems.
The computational efficiency and accuracy of the method is tested using two finite-difference solvers of the nonlinear shallow water equations on a β-plane. The method is also compared with a pure quadratic penalty approach. The updating of the Lagrangian multipliers and the penalty parameters is done using procedures suggested by Bertsekas. The method yielded satisfactory results in the conservation of the integral constraints while the additional CPU time required did not exceed 15% of the total CPU time spent on the numerical solution of the shallow water equations. The methods proved to be simple in their implementation and they have a broad scope of applicability to other problems involving nonlinear constraints; for instance, the variational nonlinear normal mode initialization.
Abstract
The Turkel–Zwas (T–Z) explicit large time-step scheme addresses the issue of fast and slow time scales in shallow-water equations by treating terms associated with fast waves on a coarser grid but to a higher accuracy than those associated with the slow-propagating Rossby waves. The T–Z scheme has been applied for solving the shallow-water equations on a fine-mesh hemispheric domain, using realistic initial conditions and an increased time step. To prevent nonlinear instability due to nonconservation of integral invariants of the shallow-water equations in long-term integrations, we enforced a posteriori their conservation. Two methods, designed to enforce a posteriori the conservation of three discretized integral invariants of the shallow-water equations, i.e., the total mass, total energy and potential enstrophy, were tested. The first method was based on an augmented Lagrangian method (Navon and de Villiers), while the second was a constraint restoration method (CRM) due to Miele et al., satisfying the requirement that the constraints be restored with the least-squares change in the field variables. The second method proved to be simpler, more efficient and far more economical with regard to CPU time, as well as easier to implement for first-time users. The CRM method has been proven to be equivalent to the Bayliss–Isaacson conservative method. The T–Z scheme with constraint restoration was run on a hemispheric domain for twenty days with no sign of impending numerical instability and with excellent conservation of the three integral invariants. Time steps approximately three times larger than allowed by the explicit CFL condition were used. The impact of the larger time step on accuracy is also discussed.
Abstract
The Turkel–Zwas (T–Z) explicit large time-step scheme addresses the issue of fast and slow time scales in shallow-water equations by treating terms associated with fast waves on a coarser grid but to a higher accuracy than those associated with the slow-propagating Rossby waves. The T–Z scheme has been applied for solving the shallow-water equations on a fine-mesh hemispheric domain, using realistic initial conditions and an increased time step. To prevent nonlinear instability due to nonconservation of integral invariants of the shallow-water equations in long-term integrations, we enforced a posteriori their conservation. Two methods, designed to enforce a posteriori the conservation of three discretized integral invariants of the shallow-water equations, i.e., the total mass, total energy and potential enstrophy, were tested. The first method was based on an augmented Lagrangian method (Navon and de Villiers), while the second was a constraint restoration method (CRM) due to Miele et al., satisfying the requirement that the constraints be restored with the least-squares change in the field variables. The second method proved to be simpler, more efficient and far more economical with regard to CPU time, as well as easier to implement for first-time users. The CRM method has been proven to be equivalent to the Bayliss–Isaacson conservative method. The T–Z scheme with constraint restoration was run on a hemispheric domain for twenty days with no sign of impending numerical instability and with excellent conservation of the three integral invariants. Time steps approximately three times larger than allowed by the explicit CFL condition were used. The impact of the larger time step on accuracy is also discussed.
Abstract
A stochastic model is presented to compute the collection kernel of cloud drops affected by turbulent air fluctuations. The model is based on stochastic differential equations with “white noise” terms representing the random turbulent displacements. The size of the random displacements is characterized by the turbulent diffusion coefficient. The associated initial boundary value problem for the partial differential equation of the Fokker-Planck type is solved numerically using finite-difference methods. For small perturbations, the collection kernel computed with the stochastic model agrees extremely well with the (deterministic) analytic expression for the collection kernel valid for non-turbulent air. The results indicate that turbulent fluctuations enhance the probability of collisions. This is particularly evident for droplets of radii less than or equal to 50 μm.
Abstract
A stochastic model is presented to compute the collection kernel of cloud drops affected by turbulent air fluctuations. The model is based on stochastic differential equations with “white noise” terms representing the random turbulent displacements. The size of the random displacements is characterized by the turbulent diffusion coefficient. The associated initial boundary value problem for the partial differential equation of the Fokker-Planck type is solved numerically using finite-difference methods. For small perturbations, the collection kernel computed with the stochastic model agrees extremely well with the (deterministic) analytic expression for the collection kernel valid for non-turbulent air. The results indicate that turbulent fluctuations enhance the probability of collisions. This is particularly evident for droplets of radii less than or equal to 50 μm.
