42 F. Hemez 5.3 The Regime of Asymptotic Convergence of Discrete Solutions Because truncation error is often the main mechanism by which discrete solutions yk n differ from the exact solution yExact, understanding its behavior is key to assess the numerical performance of simulation software. Verifying the quality of discrete solutions hinges on the regime of asymptotic convergence. Different choices of discretization variables, such as x or t, induce different types and magnitudes of error. Plotting the solution error jjyExact y k njj as a function of mesh size x gives a conceptual illustration of the main three regimes of the discretization. By definition the asymptotic regime is the region where truncation dominates the overall production of numerical error. Figure 5.2 provides a simplified illustration of these regimes. They are color-coded such that red is the regime where discretization is inappropriate, green denotes the regime of asymptotic convergence, and grey is where round-off error accumulates. Going from right (larger values of x) to left (smaller values of x) in Fig. 5.2, the first domain shown in red is where the choice of element or cell size is not even appropriate to solve the discrete equations. This would be, for example, the case when elements are too coarse to resolve important geometrical features of a contact condition between components, or a numerical stability criterion is violated. Although it could be argued that discrete solutions should not be computed in this regime, the fact is undeniable that meshes analyzed in practical situations are often significantly under-resolved. The second region shown in green is where truncation dominates the overall error: it is the regime of asymptotic convergence. Because truncation error dominates, the accuracy of a discrete solution yk can be improved simply by performing the calculation with a smaller element size. We have just described the basic principle of conducting a mesh or grid refinement study. Our conceptual illustration assumes that truncation error within the regime of asymptotic convergence is dominated by a single effect proportional to xp, which appears as a straight line of slope p on the log-log representation of Fig. 5.2. The functional form of a typical modified equation, suggested in Eq. (5.3), motivates this assumption. It is also the reason why the behavior of truncation error is usually studied by formulating a simple model such as: ". x/ D y Exact yn k . x/ ˇ x p CH:O:T:; (5.4) where "( x) denotes the difference, estimated in the sense of a user-defined normjj•jj, between the exact solution yExact of the continuous Eq. (5.1) and the discrete solution yk n( x) obtained by executing the simulation code with mesh or grid size x. The pre-factor ˇ represents a (constant) regression coefficient. The exponent p characterizes the rate at which the solution error decreases as the level of resolution is increased, that is, as x ➔0. It should match the theoretical order of 0 10-7 10-6 10-5 10-4 10-3 10-2 log(Δx) log ||yExact – y k n|| 10+1 10+0 10-1 10-2 10-3 10-4 … 10-15 10-16 Slope = –1 Slope = –½ “Stagnation” Round-off Convergence Inappropriate Fig. 5.2 The three regimes of solution error as a function of spatial discretization
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