Noninterpolatory Quadrature Formulas

1972 ◽  
Vol 9 (3) ◽  
pp. 464-475 ◽  
Author(s):  
M. P. Epstein ◽  
R. W. Hamming
Keyword(s):  
Quadrature Formulas

On Quadrature Formulas

2018 ◽  
Vol 481 (2) ◽  
pp. 136-137
Author(s):  
V. Chubarikov ◽  
Keyword(s):  
Quadrature Formulas

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

Mathematics ◽  
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.

On bi-orthogonal systems of trigonometric functions and quadrature formulas for periodic integrands

2007 ◽  
Vol 44 (4) ◽  
pp. 309-333 ◽  
Author(s):  
Ruymán Cruz-Barroso ◽  
Pablo González-Vera ◽  
Olav Njåstad
Keyword(s):  
Quadrature Formulas ◽  
Trigonometric Functions ◽  
Orthogonal Systems
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