Numerical grid discretization error

Fig. 7.3 Each pixel in the above grids marks the result of a numerical experiment using one of the four second order (in stepsize) symmetric Langevin discretization schemes. A pixel s color denotes the absolute error of q )h from an experiment conducted at the given stepsize h (horizontal) and nonlinearity parameter e (vertical) defining the potential energy in (7.13), with white pixels indicating instability. Experiments were taken over a sufficiently long time interval such that the observed errors are the result of discretization error, rather than sampling error...

Adaptive computations of nonlinear systems of reaction-diffusion equations play an increasingly important role in dynamical process simulation. The efficient adaptation of the spatial and temporal discretization is often the only way to get relevant solutions of the underlying mathematical models. The corresponding methods are essentially based on a posteriori estimates of the discretization errors. Once these errors have been computed, we are able to control time and space grids with respect to required tolerances and necessary computational work. Furthermore, the permanent assessment of the solution process allows us to clearly distinguish between numerical and modelling errors - a fact which becomes more and more important. 

The grid independence test is another important indicator that determines a successful numerical solution. Any computationally modeled real-world physical system needs to give comparable results with different mesh density or mesh size. It means that the discretization error, which is a difference between the exact numerical solution and the numerical solution due to different mesh resolution, needs to be within acceptable limits. A CFD engineer always needs to perform the grid independence test to get the correct computational results of any real-world physical system. 

M 44a] [P 40] Numerical errors which are due to discretization of the convective terms in the transport equation of the concentration fields introduce an additional, unphysical diffusion mechanism [37]. Especially for liquid-liquid mixing with characteristic diffusion constants of the order of 1CT9 m2 s 1 this so-called numerical diffusion (ND) is likely to dominate diffusive mass transfer on computational grids. 

There are couple of measures that can taken in order to minimize ND [37]. Higher order discretization schemes such as the QUICK scheme reduce the numerical errors. Furthermore, ND depends strongly on the relative orientation of flow velocity and grid cells. ND can be minimized by choosing grid cells with edges parallel to the local flow velocity. 

It is noted that the spatial-truncation error appears as an additional numerical stress, a , which depends on the grid spacing h. If the spatial discretization is p-th order accurate, then o - is of order hP. 

A consistent numerical scheme produces a system of algebraic equations which can be shown to be equivalent to the original model equations as the grid spacing tends to zero. The truncation error represents the difference between the discretized equation and the exact one. For low order finite difference methods the error is usually estimated by replacing all the nodal values in the discrete approximation by a Taylor series expansion about a single point. As a result one recovers the original differential equation plus a remainder, which represents the truncation error. 

As illustrated in the foregoing discussion, the numerical solution is expected to approach the exact solution as we use more refined discretizations. In reality, however, computer solutions may be somewhat limited in accuracy by the number of digits employed by the processor. This restriction associated with rounding to a finite number of digits results in round-off errors. These errors increase with the number of arithmetic operations required to produce a solution, hence with the number of grid points. The latter depends on the size of the problem and the degree to which the discretization is refined. The ultimate accuracy of a numerical solution can therefore be expected to be achieved in a trade-off between truncation errors and round-off errors. The effect of... 

After specification of the required tolerances for the errors of the space discretization (tolx) and the time discretization (to/t), respectively, the code can directly be used to solve the problem as the Jacobian matrix of the residual is approximated internally by a numerical differentiation scheme. In the standard case, the numerical solution is available at the internally selected integration points and on the associated computational grid. There are some optional parameters which can be set to optimize the performance for the problem at hand, e.g. special modes for linear algebra manipulation and Jacobian approximation. Furthermore, there are three features of the package which are worth to be discussed in more detail. 

Finite-difference methods operating on a grid consisting of equidistant points ( Xi, Xi = ih + Xq) are known to be one of the most accurate techniques available [496]. Additionally, on an equidistant grid all discretized operators appear in a simple form. The uniform step size h allows us to use the Richardson extrapolation method [494,497] for the control of the numerical truncation error. Many methods are available for the discretization of differential equations on equidistant grids and for the integration (quadrature) of functions needed for the calculation of expectation values. 

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