An optimized implementation of the Newmark/Newton-Raphson algorithm for the time integration of non-linear problems

A condition expressed in Eq. (7) is given which, with one simplifying regularity condition, ensures that the policy-improvement algorithm is equivalent to application of the Newton–Raphson algorithm to an optimality condition. It is shown that this condition covers the two known cases of such equivalence, and another example is noted. The condition is believed to be necessary to within transformations of the problem, but this has not been proved.

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Collocation Methods for High-Order Well-Balanced Methods for Systems of Balance Laws

Mathematics ◽

10.3390/math9151799 ◽

2021 ◽

Vol 9 (15) ◽

pp. 1799

Author(s):

Irene Gómez-Bueno ◽

Manuel Jesús Castro Díaz ◽

Carlos Parés ◽

Giovanni Russo

Keyword(s):

High Order ◽

Collocation Methods ◽

Quadrature Formulas ◽

Balance Laws ◽

Source Terms ◽

Time Step ◽

Systems Of Balance Laws ◽

Reconstruction Operator ◽

Non Linear ◽

Linear Problems

In some previous works, two of the authors introduced a technique to design high-order numerical methods for one-dimensional balance laws that preserve all their stationary solutions. The basis of these methods is a well-balanced reconstruction operator. Moreover, they introduced a procedure to modify any standard reconstruction operator, like MUSCL, ENO, CWENO, etc., in order to be well-balanced. This strategy involves a non-linear problem at every cell at every time step that consists in finding the stationary solution whose average is the given cell value. In a recent paper, a fully well-balanced method is presented where the non-linear problems to be solved in the reconstruction procedure are interpreted as control problems. The goal of this paper is to introduce a new technique to solve these local non-linear problems based on the application of the collocation RK methods. Special care is put to analyze the effects of computing the averages and the source terms using quadrature formulas. A general technique which allows us to deal with resonant problems is also introduced. To check the efficiency of the methods and their well-balance property, they have been applied to a number of tests, ranging from easy academic systems of balance laws consisting of Burgers equation with some non-linear source terms to the shallow water equations—without and with Manning friction—or Euler equations of gas dynamics with gravity effects.

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Oppositely converging Newton-Raphson method for non-linear equilibrium problems

International Journal for Numerical Methods in Engineering ◽

10.1002/nme.2574 ◽

2009 ◽

Vol 79 (3) ◽

pp. 375-378 ◽

Cited By ~ 2

Author(s):

Isaac Fried

Keyword(s):

Equilibrium Problems ◽

Non Linear ◽

Newton Raphson ◽

Raphson Method

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An efficient algorithm for estimating the parameters of superimposed exponential signals in multiplicative and additive noise

International Journal of Applied Mathematics and Computer Science ◽

10.2478/amcs-2013-0010 ◽

2013 ◽

Vol 23 (1) ◽

pp. 117-129 ◽

Cited By ~ 2

Author(s):

Jiawen Bian ◽

Huiming Peng ◽

Jing Xing ◽

Zhihui Liu ◽

Hongwei Li

Keyword(s):

Parameter Estimation ◽

Convergence Rate ◽

Efficient Algorithm ◽

Numerical Experiments ◽

Additive Noise ◽

Mean Squared Errors ◽

Least Squares Estimators ◽

The Mean ◽

Newton Raphson ◽

Raphson Algorithm

This paper considers parameter estimation of superimposed exponential signals in multiplicative and additive noise which are all independent and identically distributed. A modified Newton-Raphson algorithm is used to estimate the frequencies of the considered model, which is further used to estimate other linear parameters. It is proved that the modified Newton- Raphson algorithm is robust and the corresponding estimators of frequencies attain the same convergence rate with Least Squares Estimators (LSEs) under the same noise conditions, but it outperforms LSEs in terms of the mean squared errors. Finally, the effectiveness of the algorithm is verified by some numerical experiments.

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