Physical Sciences and Mathematics Commons^™

Accuracy, Resolution And Stability Properties Of A Modified Chebyshev Method, Jodi Mead, Rosemary A. Renaut

Jodi Mead

While the Chebyshev pseudospectral method provides a spectrally accurate method, integration of partial differential equations with spatial derivatives of order M requires time steps of approximately O(N−2M) for stable explicit solvers. Theoretically, time steps may be increased to O(N−M) with the use of a parameter, α-dependent mapped method introduced by Kosloff and Tal-Ezer [ J. Comput. Phys., 104 (1993), pp. 457–469]. Our analysis focuses on the utilization of this method for reasonable practical choices for N, namely N ≲ 30, as may be needed for two- or three dimensional modeling. Results presented confirm that spectral accuracy with increasing N is …

The Shallow Water Equations In Lagrangian Coordinates, J. L. Mead

Jodi Mead

Recent advances in the collection of Lagrangian data from the ocean and results about the well-posedness of the primitive equations have led to a renewed interest in solving flow equations in Lagrangian coordinates. We do not take the view that solving in Lagrangian coordinates equates to solving on a moving grid that can become twisted or distorted. Rather, the grid in Lagrangian coordinates represents the initial position of particles, and it does not change with time. However, using Lagrangian coordinates results in solving a highly nonlinear partial differential equation. The nonlinearity is mainly due to the Jacobian of the coordinate …

Towards Regional Assimilation Of Lagrangian Data: The Lagrangian Form Of The Shallow Water Reduced Gravity Model And Its Inverse, J. L. Mead, A. F. Bennett

Jodi Mead

Variational data assimilation for Lagrangian geophysical fluid dynamics is introduced. Lagrangian coordinates add numerical difficulties into an already difficult subject, but also offer certain distinct advantages over Eulerian coordinates. First, float position and depth are defined by linear measurement functionals. Second, Lagrangian or ‘comoving’ open domains are conveniently expressed in Lagrangian coordinates. The attraction of such open domains is that they lead to well-posed prediction problems [Bennett and Chua (1999)] and hence efficient inversion algorithms. Eulerian and Lagrangian solutions of the inviscid forward problem in a doubly periodic domain, with North Atlantic mesoscales, are compared and found to be in …

An Iterated Pseudospectral Method For Functional Partial Differential Equations, J. Mead, B. Zubik-Kowal

Jodi Mead

Chebyshev pseudospectral spatial discretization preconditioned by the Kosloff and Tal-Ezer transformation [10] is applied to hyperbolic and parabolic functional equations. A Jacobi waveform relaxation method is then applied to the resulting semi-discrete functional systems, and the result is a simple system of ordinary differential equations d/dtUk+1(t) = MαUk+1(t)+f(t,U kt). Here Mα is a diagonal matrix, k is the index of waveform relaxation iterations, U kt is a functional argument computed from the previous iterate and the function f, like the matrix Mα, depends on the process of semi-discretization. This waveform relaxation splitting has the advantage of straight forward, direct application …

Assimilation Of Simulated Float Data In Lagrangian Coordinates, J. L. Mead

Jodi Mead

We implement an approach for the accurate assimilation of Lagrangian data into regional general ocean circulation models. The forward model is expressed in Lagrangian coordinates and simulated float data are incorporated into the model via four dimensional variational data assimilation. We show that forward solutions computed in Lagrangian coordinates are reliable for time periods of up to 100 days with phase speeds of 1 m/s and deformation radius of 35 km. The position and depth of simulated floats are assimilated into the viscous, Lagrangian shallow water equations. The weights for the errors in the model and data are varied and …

Least Squares Problems With Inequality Constraints As Quadratic Constraints, Jodi Mead, Rosemary A. Renaut

Jodi Mead

Linear least squares problems with box constraints are commonly solved to find model parameters within bounds based on physical considerations. Common algorithms include Bounded Variable Least Squares (BVLS) and the Matlab function lsqlin. Here, the goal is to find solutions to ill-posed inverse problems that lie within box constraints. To do this, we formulate the box constraints as quadratic constraints, and solve the corresponding unconstrained regularized least squares problem. Using box constraints as quadratic constraints is an efficient approach because the optimization problem has a closed form solution.

The effectiveness of the proposed algorithm is investigated through solving three …

A Newton Root-Finding Algorithm For Estimating The Regularization Parameter For Solving Ill-Conditioned Least Squares Problems, Jodi Mead, Rosemary Renaut

Jodi Mead

We discuss the solution of numerically ill-posed overdetermined systems of equations using Tikhonov a-priori-based regularization. When the noise distribution on the measured data is available to appropriately weight the fidelity term, and the regularization is assumed to be weighted by inverse covariance information on the model parameters, the underlying cost functional becomes a random variable that follows a X2 distribution. The regularization parameter can then be found so that the optimal cost functional has this property. Under this premise a scalar Newton root-finding algorithm for obtaining the regularization parameter is presented. The algorithm, which uses the singular value decomposition of …