scholarly journals Research Article. Geodesic equations and their numerical solutions in geodetic and Cartesian coordinates on an oblate spheroid

2017 ◽  
Vol 7 (1) ◽  
Author(s):  
G. Panou ◽  
R. Korakitis
Keyword(s):  
Initial Value Problem ◽  
Numerical Solutions ◽  
Oblate Spheroid ◽  
Initial Value ◽  
Data Set ◽  
Cartesian Coordinates ◽  
Geodesic Equations ◽  
Research Article ◽  
Fast Solution ◽  
Geodesic Problem

AbstractThe direct geodesic problem on an oblate spheroid is described as an initial value problem and is solved numerically using both geodetic and Cartesian coordinates. The geodesic equations are formulated by means of the theory of differential geometry. The initial value problem under consideration is reduced to a system of first-order ordinary differential equations, which is solved using a numerical method. The solution provides the coordinates and the azimuths at any point along the geodesic. The Clairaut constant is not used for the solution but it is computed, allowing to check the precision of the method. An extensive data set of geodesics is used, in order to evaluate the performance of the method in each coordinate system. The results for the direct geodesic problem are validated by comparison to Karney’s method. We conclude that a complete, stable, precise, accurate and fast solution of the problem in Cartesian coordinates is accomplished.

scholarly journals Geodesic equations and their numerical solution in Cartesian coordinates on a triaxial ellipsoid

2019 ◽  
Vol 9 (1) ◽  
pp. 1-12 ◽  
Author(s):  
G. Panou ◽  
R. Korakitis
Keyword(s):  
Numerical Solution ◽  
Numerical Method ◽  
Oblate Spheroid ◽  
Triaxial Ellipsoid ◽  
Data Set ◽  
Cartesian Coordinates ◽  
Geodesic Equations ◽  
Ellipsoidal Coordinates ◽  
Extensive Test ◽  
Geodesic Problem

Abstract In this work, the geodesic equations and their numerical solution in Cartesian coordinates on an oblate spheroid, presented by Panou and Korakitis (2017), are generalized on a triaxial ellipsoid. A new exact analytical method and a new numerical method of converting Cartesian to ellipsoidal coordinates of a point on a triaxial ellipsoid are presented. An extensive test set for the coordinate conversion is used, in order to evaluate the performance of the two methods. The direct geodesic problem on a triaxial ellipsoid is described as an initial value problem and is solved numerically in Cartesian coordinates. The solution provides the Cartesian coordinates and the angle between the line of constant λ and the geodesic, at any point along the geodesic. Also, the Liouville constant is computed at any point along the geodesic, allowing to check the precision of the method. An extensive data set of geodesics is used, in order to demonstrate the validity of the numerical method for the geodesic problem. We conclude that a complete, stable and precise solution of the problem is accomplished.

The direct geodesic problem and an approximate analytical solution in Cartesian coordinates on a triaxial ellipsoid

2020 ◽  
Vol 14 (2) ◽  
pp. 205-213
Author(s):  
G. Panou ◽  
R. Korakitis
Keyword(s):  
Numerical Method ◽  
Analytical Method ◽  
Taylor Series Expansion ◽  
Oblate Spheroid ◽  
Accurate Solution ◽  
Triaxial Ellipsoid ◽  
Data Set ◽  
Cartesian Coordinates ◽  
Similar Accuracy ◽  
Geodesic Problem

AbstractIn this work, the direct geodesic problem in Cartesian coordinates on a triaxial ellipsoid is solved by an approximate analytical method. The parametric coordinates are used and the parametric to Cartesian coordinates conversion and vice versa are presented. The geodesic equations on a triaxial ellipsoid in Cartesian coordinates are solved using a Taylor series expansion. The solution provides the Cartesian coordinates and the angle between the line of constant v and the geodesic at the end point. An extensive data set of geodesics, previously studied with a numerical method, is used in order to validate the presented analytical method in terms of stability, accuracy and execution time. We conclude that the presented method is suitable for a triaxial ellipsoid with small eccentricities and an accurate solution is obtained. At a similar accuracy level, this method is about thirty times faster than the corresponding numerical method. Finally, the presented method can also be applied in the degenerate case of an oblate spheroid, which is extensively used in geodesy.

Operational predictability of winter blocking at ECMWF: an update

1995 ◽  
Vol 13 (3) ◽  
pp. 305-317 ◽  
Author(s):  
S. Tibaldi ◽  
P. Ruti ◽  
E. Tosi ◽  
M. Maruca
Keyword(s):  
Initial Value Problem ◽  
Weather Forecast ◽  
Model Resolution ◽  
Initial Value ◽  
Atlantic Region ◽  
Data Set ◽  
Medium Range Weather Forecast ◽  
Medium Range ◽  
Limited Length ◽  
Diagnostic Work

Abstract. Seven winters of analyses and forecasts from the operational archives of the European Centre for Medium Range Weather Forecast had been previously analyzed to assess the performance of the model in forecasting blocking events. This work updates some of this previous diagnostic work to the last five winters, from 1987/88 to 1991/92. The data set therefore covers all winter seasons (DJF) from 1980/81 to 1991/92, and consists of daily northern hemisphere 500 hPa geopotential height analyses and of the ten corresponding forecasts verifying on the same day ("Lorenz data"). Local blocking and sector blocking have been defined, using different modifications of the original Lejenas and Økland index. The comparison between the first seven and the last five winters, within the restrictions imposed by limited length of the data set, suggests a much improved situation as far as model climatology of blocking is concerned, especially over the Euro-Atlantic region. Operational predictability of blocking as an initial value problem is also shown to be measurably improved, in both Atlantic and Pacific sectors. All such improvements are shown to have taken place together with a considerable reduction of the model systematic error. Nevertheless, forecasting blocking in the medium range remains a difficult task for the model. More work is needed to understand whether the improvements are to be ascribed to the increased model resolution or to better physical parametrisations.

Numerical solutions for fractional initial value problems of distributed-order

2021 ◽  
pp. 2150096
Author(s):  
M. A. Abdelkawy
Keyword(s):  
Fractional Order ◽  
Initial Value Problem ◽  
Numerical Solutions ◽  
Jacobi Polynomials ◽  
Algebraic Equations ◽  
Initial Value ◽  
Orthogonal Functions ◽  
Jacobi Functions ◽  
Distributed Order ◽  
Numerical Approaches

This paper addresses spectral collocation techniques to treat with the fractional initial value problem of distributed-order. We introduce three algorithms based on shifted fractional order Jacobi orthogonal functions outputted by Jacobi polynomials. The shifted fractional order Jacobi–Gauss–Radau collocation method is developed for approximating the fractional initial value problem of distributed-order. The principal target in our techniques is to transform the fractional initial value problem of distributed-order to a system of algebraic equations. Some numerical examples are given to test the accuracy and applicability of our technique. It is known that the accuracy of numerical approaches for nonsmooth solution is deteriorated. Employing fractional order Jacobi functions instead of the classical Jacobi stopped this deterioration.

scholarly journals Numerical solutions of nonstandard first order initial value problems

1992 ◽  
Vol 5 (1) ◽  
pp. 69-82 ◽  
Author(s):  
M. Venkatesulu ◽  
P. D. N. Srinivasu
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Unique Solution ◽  
Initial Value Problem ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Initial Value ◽  
First Order ◽  
Physical Phenomena

Differential equations of the form y′=f(t,y,y′), where f is not necessarily linear in its arguments, represent certain physical phenomena and are known for quite some time. The well known Clairut's and Chrystal's equations fall into this category. Earlier, we established the existence of a (unique) solution of the nonstandard initial value problem (NSTD IV P) y′=f(t,y,y′), y(t0)=y0 under certain natural hypotheses on f. In this paper we present some first order convergent numerical methods for finding the approximate solutions of the NST D I V Ps.

The development of travelling waves in a simple isothermal chemical system - I. Quadratic autocatalysis with linear decay

1989 ◽  
Vol 424 (1866) ◽  
pp. 187-209 ◽  
Keyword(s):  
Initial Value Problem ◽  
Numerical Solutions ◽  
Travelling Waves ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Chemical System ◽  
Length Scale ◽  
Diffusion Front ◽  
Initial Value ◽  
First Case

The possibility of travelling reaction–diffusion waves developing in the chemical system governed by the quadratic autocatalytic or branching reaction A + B → 2B (rate k 1 ab ) coupled with the decay or termination step B → C (rate k 2 b ) is examined. Two simple solutions are obtained first, namely the well-stirred analogue of the spatially inhomogeneous problem and the solution for small input of the reactant B. Both of these indicate that the criterion for the existence of a travelling wave is that k 2 < k 1 a 0 , where a 0 is the initial concentration of reactant A. The equations governing the fully developed travelling waves are then discussed and it is shown that these possess a solution only if this criterion is satisfied, i. e. only if k = k 2 / k 1 a 0 < 1. Further properties of these waves are also established and, in particular, it is shown that the concentration of A increases monotonically from its fully reacted state at the rear of the wave to its unreacted state at the front, while the concentration of B has a single hump form. Numerical solutions of the full initial value problem are then obtained and these do confirm that travelling waves are possible only if k < 1 and suggest that, when this condition holds, these waves travel with the uniform speed v 0 = 2√ (1 – k ). This last result is established by a large time analysis of the full initial value problem that reveals that ahead of the reaction–diffusion front is a very weak diffusion-controlled region into which an exponentially small amount of B must diffuse before the reaction can be initiated. Finally, the behaviour of the travelling waves in the two asymptotic limits k → 0 and k → 1 are treated. In the first case the solution approaches that for the previously discussed k = 0 case on the length scale associated with the reaction–diffusion front, with the difference being seen on a much longer, O ( k –1 ), length scale. In the latter case we find that the concentration of A is 1 + O (1 – k ) and that of B is O ((1 – k ) 2 ), with the thickness of the reaction–diffusion front being of O ((1 – k ) ½ ).

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