On the relation between order of accuracy, convergence rate and spectral slope for linear numerical methods applied to multiscale problems

2008 ◽  
Vol 56 (8) ◽  
pp. 1297-1303 ◽  
Author(s):  
D. Holdaway ◽  
J. Thuburn ◽  
N. Wood
Keyword(s):  
Numerical Methods ◽  
Convergence Rate ◽  
Spectral Slope ◽  
Order Of Accuracy ◽  
Multiscale Problems

Numerical methods with a high order of accuracy applied in the quantum system

1996 ◽  
Vol 104 (6) ◽  
pp. 2275-2286 ◽  
Author(s):  
Wusheng Zhu ◽  
Xinsheng Zhao ◽  
Youqi Tang
Keyword(s):  
Numerical Methods ◽  
Quantum System ◽  
High Order ◽  
Order Of Accuracy ◽  
High Order Of Accuracy

scholarly journals Methods for solving the initial value problem with a two-sided estimate of the local error

2021 ◽  
pp. 88-92
Author(s):  
Yaroslav Pelekh ◽  
Andrii Kunynets ◽  
Halyna Beregova ◽  
Tatiana Magerovska
Keyword(s):  
Differential Equation ◽  
Numerical Methods ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problem ◽  
Local Error ◽  
Initial Value ◽  
Order Of Accuracy ◽  
Embedded Methods ◽  
The Right

Numerical methods for solving the initial value problem for ordinary differential equations are proposed. Embedded methods of order of accuracy 2(1), 3(2) and 4(3) are constructed. To estimate the local error, two-sided calculation formulas were used, which give estimates of the main terms of the error without additional calculations of the right-hand side of the differential equation, which favorably distinguishes them from traditional two-sided methods of the Runge- Kutta type.

Functional-Discrete Method with a High Order of Accuracy for the Eigenvalue Transmission Problem

2004 ◽  
Vol 4 (3) ◽  
pp. 324-349 ◽  
Author(s):  
V.L. Makarov ◽  
N.O. Rossokhata ◽  
B.I. Bandurskii
Keyword(s):  
Convergence Rate ◽  
Original Problem ◽  
Basic Problem ◽  
Ordinal Number ◽  
High Order ◽  
Transmission Problem ◽  
Order Of Accuracy ◽  
Discrete Method ◽  
Numerical Examples ◽  
High Order Of Accuracy

AbstractWe develop a functional-discrete method with a high order of accuracy to find a numerical solution of an eigenvalue transmission problem. It allows to approximate the trial eigenvalue with any desired accuracy. This approach has no restriction on the number of eigenvalues, an approximation to which can be found. The convergence rate is proved as in the case of the geometric series. It is shown that depending on the data of the original problem, two kinds of eigenvalue sequences may exist. For the first one, the convergence rate increases as the ordinal number of the trial eigenvalue increases. For the second one, the convergence rate is the same for all eigenvalues and does not depend on the ordinal number of the trial eigenvalue. Based on the asymptotic behavior of the eigenvalues of the basic problem and the functional-discrete method, a qualitative result on the arrangement of eigenvalues of the original problem is established. A number of numerical examples are given to support the theory.

scholarly journals Investigation of Finite-Difference Schemes for the Numerical Solution of a Fractional Nonlinear Equation

2021 ◽  
Vol 6 (1) ◽  
pp. 23
Author(s):  
Dmitriy Tverdyi ◽  
Roman Parovik
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Difference Scheme ◽  
Numerical Solution ◽  
Riccati Equation ◽  
Finite Difference Scheme ◽  
Stability And Convergence ◽  
Computational Accuracy ◽  
Order Of Accuracy ◽  
First Order

The article discusses different schemes for the numerical solution of the fractional Riccati equation with variable coefficients and variable memory, where the fractional derivative is understood in the sense of Gerasimov-Caputo. For a nonlinear fractional equation, in the general case, theorems of approximation, stability, and convergence of a nonlocal implicit finite difference scheme (IFDS) are proved. For IFDS, it is shown that the scheme converges with the order corresponding to the estimate for approximating the Gerasimov-Caputo fractional operator. The IFDS scheme is solved by the modified Newton’s method (MNM), for which it is shown that the method is locally stable and converges with the first order of accuracy. In the case of the fractional Riccati equation, approximation, stability, and convergence theorems are proved for a nonlocal explicit finite difference scheme (EFDS). It is shown that EFDS conditionally converges with the first order of accuracy. On specific test examples, the computational accuracy of numerical methods was estimated according to Runge’s rule and compared with the exact solution. It is shown that the order of computational accuracy of numerical methods tends to the theoretical order of accuracy with increasing nodes of the computational grid.

scholarly journals ROBUST NOVEL HIGH-ORDER ACCURATE NUMERICAL METHODS FOR SINGULARLY PERTURBED CONVECTION‐DIFFUSION PROBLEMS

2005 ◽  
Vol 10 (4) ◽  
pp. 393-412 ◽  
Author(s):  
G. I. Shishkin
Keyword(s):  
Numerical Methods ◽  
Order Of Convergence ◽  
Diffusion Problem ◽  
Singularly Perturbed ◽  
Order Of Accuracy ◽  
Convection Diffusion ◽  
Expansion Technique ◽  
Uniformly Convergent ◽  
Problem Data ◽  
Diffusion Problems

For singularly perturbed boundary value problems, numerical methods convergent ϵ‐uniformly have the low accuracy. So, for parabolic convection‐diffusion problem the order of convergence does not exceed one even if the problem data are sufficiently smooth. However, already for piecewise smooth initial data this order is not higher than 1/2. For problems of such type, using newly developed methods such as the method based on the asymptotic expansion technique and the method of the additive splitting of singularities, we construct ϵ‐uniformly convergent schemes with improved order of accuracy. Straipsnyje nagrinejami nedidelio tikslumo ϵ‐tolygiai konvertuojantys skaitmeniniai metodai, singuliariai sutrikdytiems kraštiniams uždaviniams. Paraboliniam konvekcijos‐difuzijos uždaviniui konvergavimo eile neviršija vienos antrosios, jeigu uždavinio duomenys yra pakankamai glodūs. Tačiau trūkiems pradiniams duomenims eile yra ne aukštesne už 2−1. Šio tipo uždaviniams, naudojant naujai išvestus metodus, darbe sukonstruotos ϵ‐tolygiai konvertuojančios schemos aukštesniu tikslumu.

Numerical Methods

2019 ◽  
Author(s):  
Rajesh Kumar Gupta
Keyword(s):  
Numerical Methods
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