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FINITE-DIFFERENCE FORMULATION
Published in Wenquan Sui, Time-Domain Computer Analysis of Nonlinear Hybrid Systems, 2018
where fC'(x0) is the approximated derivative of f(x) at x0, using central finitedifference. As shown in the above equations, forward, backward or central difference can be implemented for a numerical approximation of the first-order derivative for the single-variable function f(x). The criterion for choosing which method to use is highly dependent upon the accuracy requirement, stability consideration and available computing resources. There are two major contributors of numerical error; one is truncation error and the other is computer round-off error. Round-off error is directly related to the computer word length, while the truncation error is algorithm dependent and will be discussed briefly in the following. From (4.4), it is easy to see that the forward finite-difference approximation of the first derivative is expressed as
Vectors, Matrices, and Linear Systems
Published in Chee Khiang Pang, Frank L. Lewis, Tong Heng Lee, Zhao Yang Dong, Intelligent Diagnosis and Prognosis of Industrial Networked Systems, 2017
Chee Khiang Pang, Frank L. Lewis, Tong Heng Lee, Zhao Yang Dong
The condition number of A or cond(A) is a relative measure of how close A is to rank deficiency, i.e., how close A is to being singular or non-invertible, and can be interpreted as a measure of how much numerical error is likely to be introduced by computations involving A. The condition number is defined as cond(A)=σ1σn, i.e., the ratio of the largest singular value σ1 to the smallest singular value σn. As such, it is obvious that singular matrices have at least one zero singular value, which leads to an infinite or undefined condition number. A matrix A with a large cond(A) is said to be ill-conditioned or stiff, and the integrity of numerical solutions obtained when handling these matrices should be confronted. On the other hand, a matrix A with a small cond(A) close to unity is desirable, and is said to be well-conditioned. Matrices with a small condition number near unity are usually balanced, with small variations amongst entries in the matrix or link gains as discussed in the previous section.
Numerical Analysis
Published in Erchin Serpedin, Thomas Chen, Dinesh Rajan, Mathematical Foundations for SIGNAL PROCESSING, COMMUNICATIONS, AND NETWORKING, 2012
Numerical error is the difference between the true value and the computed value of a quantity. The significance of error depends on the magnitude of the true value. We define absolute error and relative error to represent this notion formally: () AbsoluteError=a^−a, () RelativeError=a^−aa,
CFD-based predictions of hydrodynamic forces in ship-tug boat interactions
Published in Ships and Offshore Structures, 2019
Since the viscous flow surrounding the KVLCC2 and the tug boat is more complicated than the case with a mono hull, special attentions should be paid to the accuracy in the numerical simulations, in order to produce reliable results. That is, the numerical error δRE and uncertainty USN in the simulation should be carefully estimated before any systematic computation. Assuming that the round-off error is negligible, the main contribution to the numerical error comes from the discretization of the mathematical equations in space and/or in time. For a steady simulation as presented here, only the grid discretization error needs to be considered and can be estimated from a grid convergence study.
Computational fluid dynamic (CFD) modelling in anaerobic digestion: General application and recent advances
Published in Critical Reviews in Environmental Science and Technology, 2018
Constanza Sadino-Riquelme, Robert E. Hayes, David Jeison, Andrés Donoso-Bravo
Just like any other model, CFD application will comprise different types of error and all the possible error sources should be considered. The error sources include model errors, discretization errors, numerical errors and programming errors. Model error basically refers to whether or not the model equations are in fact a good representation of the physics. Discretization error depends on the mesh used and the scheme applied for the discretization in time and space of the conservation equations. Numerical error relates to errors caused be the numerical method chosen and rounding error in the computer. Programming error simply means incorrect input by the programmer.