Figure 64. Dependence of the relative errorEA;rel on parameterssvrec and acoa.
over uniform grids with the same CPU time. This error is de…ned by EA;rel = EAjadaptive
EAjuniform; (179)
where the errors on the right hand side combine the global, normalised errors of the surface elevation and the velocity vector for adaptive and uniform meshes:
EA= X2
i=1
[EA( )]2t
i + [EA(v)]2t
i ; (180)
witht1 = 3000 sand t2 = 4000 s. The current test problem is solved repeatedly on solution-adapted meshes governed by the velocity reconstruction error with more than 30 combinations of svrec and acoa; this yields EAjadaptive. Based on the CPU time required by each adaptive simulation, the corresponding error EAjuniform is interpolated by …tting a power law on the relationship EA $ tCPU of uniform grids. The dependence is presented in Figure 64. Thus, in this problem, the highest accuracy gain of solution-adaptivity is 35% compared to uniform grids with the same CPU time. The optimum is obtained with svrec = 0:15: : :0:2, whereas the other parameter, acoa may be chosen from a wide range without noticeably a¤ecting the error. A higher limit to the number of cells combined with a higher adaptation sensitivity tends to re…ne the mesh more uniformly, which brings the performance close to that obtained with regular grids. By allowing deeper levels of adaptation, the resulting timestep penalty increases the total number of iterations.
5.7. CONCLUDING REMARKS 111 quantitatively the better performance of adaptive schemes in comparison with uniform grids in terms of cell count. The CFL penalty in small cells is somewhat relieved by local time stepping, which helps to keep the performance gain also in terms of the CPU time. The re…nement procedure is remarkably robust and general, as only minimal and predictable …ne-tuning of the adaptation parameters was necessary to accommodate a steady-state as well as an unsteady benchmark problem.
Solution-adaptation alleviates the need for manual, a priori de…nition of the initial mesh. For example, ‡ow in the direction of high depth gradient is mon-itored by the error indicator based on divergence, whereas the perpendicular ‡ow is tracked based on circulation. Consequently, a re…nement criterion linked to the topography can be reformulated by using an error indicator linked to the ‡ow solu-tion. The same applies to boundary re…nement, as wall shear produces vorticity hence activates the circulation-based error indicator. In spite of the automatism o¤ered by solution-adaptation, care must be taken when setting up the initial mesh. A minimal resolution is in fact necessary for small-scale bed features and thin vegetation patches to come within the scope of the ‡ow solver. The error estimator will then re…ne it further if deemed necessary from the point of view of solution accuracy.
Even with linearly varying conserved variables, re…nement improves the accur-acy of those terms in the governing equations which are a nonlinear function ofu.
Hence the bene…t of re…nement is not only a better approximation of the sub-cell distribution but also the introduction of additional Gauss quadrature points to evaluate path and surface integrals of the nonlinear momentum equation.
The selection of the adaptation indicator depends on how we de…ne the ac-curacy of the solution. For the two solution-adaptive wind-driven ‡ow problems, the indicator based on the velocity reconstruction error (“ad, v-rec”) performed reliably in all tests. The other indicator that proved equally robust overall was
“ad, v-rec”that is based on the combination of circulation, divergence and velocity reconstruction error.
The formal second-order accuracy is con…rmed only for the surface elevation and in the simulation of ‡ow in rectangular basins, whereas the order of accuracy is reduced to between 1 and 1.5 in problems with circular boundaries. The test cases involving circular boundaries revealed that the Cartesian MUSCL scheme does not handle arbitrarily oriented boundaries well because excessive spurious drag is produced by the stepped representation of such boundaries. This leads to an order of accuracy with respect to the velocity lower by about 0.5 than with respect to the elevation. This damaging e¤ect is mitigated by extra re…nement,
though at a higher total computational cost. Nevertheless, if the target of the modelling is a lake with a shallow, perhaps vegetated coastal littoral zone with slow currents, the spurious drag is expected to manifest itself less severely than in this analytical benchmark problem.
Chapter VI
APPLICATION TO LAKE NEUSIEDL
In this chapter, the physical validity of the model and its applicability to real con-ditions is demonstrated with case studies of Lake Neusiedl. Its patchy reed cover permits the veri…cation of a nested internal boundary layer model; the single- and multiple-IBL model are studied for parameter sensitivity, then calibrated against simultaneous wind measurements. Parameter sensitivity is also analysed for the Grant-Madsen wave-current interaction model. The wave formulae are veri…ed with wave data collected in the bay of Fert½orákos, whereas the e¤ect of the wave enhancement of bottom shear stress on the circulation is studied numerically in a simpli…ed representation of the bay. To close the chapter, the adaptive ‡ow model is applied to the lake to determine circulations due to the prevailing storms in the whole lake and in two bays surrounded by reed. Static and solution-driven mesh adaptivity is used to accelerate convergence to steady state, localise mesh resolu-tion to the area of interest and to reallocate cells during unsteady simularesolu-tions.