### Table 1 Numerical results for Example 1. Discrete delta function immersed interface method n

1994

"... In PAGE 21: ... We see that our method accurately gives the jump in the normal direction while the discrete delta function approach smears the jump, resulting in rst order accuracy. Table1 shows the results of a grid re nement study. The maximum error over all grid... ..."

Cited by 85

### Table 1: Analytical values of Re( ) for so- lution modes from the smoothed and ex- act delta{function problems along with the computed decay rates (N = 128). We have implemented the Immersed Boundary Method and computed the rate of decay of the os- cillations for a at ber given a small perturbation from equilibrium. This rate corresponds to the most slowly decaying modes ( = 1), since the higher wavenumber modes decay most quickly. From the table, it is clear that the behaviour of the lowest wavenumber modes is captured quite accurately by the numerical scheme.

"... In PAGE 4: ... From the table, it is clear that the behaviour of the lowest wavenumber modes is captured quite accurately by the numerical scheme. The nal column in Table1 presents the values of based on the results in [11] for the exact delta func- tion (that is, without smoothing). It is clear that the introduction of smoothing has a considerable e ect on the solution modes.... ..."

### Table 5: E ectiveness Factors for (1.4) using Quadrature for both problem (1.3) and problem (1.4). The paper also introduces a more accurate method for calculating the e ectivness factor numerically. References [1] R. E. Bellman, R. E. Kalaba, Quasilinearization and Nonlinear Boundary{Value Prob- lems, Elsevier Publishing Company, New York, 1965. [2] K. W. Chang, F. A. Howes, Nonlinear Singular Perturbation Phenomena: Theory and Applications, Springer{Verlag, New York, 1984. [3] E. P. Doolan, J. J. H. Miller, W. H. A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, Dublin, 1980. [4] P.A. Farrell, A. Hegarty, On the determination of the order of uniform convergence. In: Proc. of 13th IMACS World Congress (J. J. H. Miller, R. Vichnevetsky, eds.), Dublin, 1991, 501{502.

1993

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### Table 2.1: Numerical results for Example 2.1 Discrete delta function Immersed interface method n

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### Table 1: Boundary conditions

in SUMMARY

"... In PAGE 5: ... The narrowness of the Gaussian pro#0Cle has a relevant in#0Duence on the calculated #0Dame length, so that its parameters have to be determined appropriately #5B19#5D. The boundary conditions are summarized in Table1 . Finally, we note that the use of the de#0Cnition of the vorticity #281#29 for the vorticity outlet boundary condition does not yield any relevantchanges in the computed solution.... In PAGE 5: ... 3. GENERAL SOLUTION ALGORITHM The partial di#0Berential equations #282#29 together with the boundary conditions #28see Table1 #29 are discretized on a two dimensional tensor product grid. A solution is #0Crst obtained on an initial coarse grid.... In PAGE 6: ...i#0Berence expressions. Di#0Busion and source terms are evaluated using centered di#0Berences. We adopt a monotonicity preserving upwind scheme for the convective terms #28see #5B20, p. 304#5D#29, for instance, v r @S @r = maxf#28v r #29 i, 1 2 ; 0g S i , S i,1 r i , r i,1 , maxf,#28v r #29 i+ 1 2 ; 0g S i+1 , S i r i+1 , r i : #283#29 The boundary conditions given in Table1 involve only zero or #0Crst order derivatives. For the latter terms, #0Crst order back or forward di#0Berences can be used, except for two boundary conditions which require a more accurate treatment.... In PAGE 7: ... By comparing our numerical solutions with a primitivevariable solution of the same problem #5B19#5D, we found that these two boundary conditions exerted a strong in#0Duence on the overall accuracy of the numerical solution. The discretization of the partial di#0Berential equations #282#29 together with the boundary conditions #28 Table1 #29 yields a set of algebraic equations of the form F #28U#29 = 0, which is solved using a damped Newton method J#28U n #29#01U n = ,#15 n F #28U n #29; n =0;1;:::; #285#29 with convergence tolerance k#01U n k S #3C 10 ,5 . The Jacobian matrix J#28U n #29 is computed numerically using vector function evaluations and the grid nodes are split into nine independent groups which are perturbed simultaneously #28see #5B2#5D for more details#29.... ..."

### TABLE 2 Gaussian Quadrature Method

### Table 4: Convergence of the quadrature method for g1 and = 0.5 at P =(1,0,0)

"... In PAGE 31: ...compute the error jUNl(P ) ? UNl+1(P )j and the corresponding convergence rates Nl = log jUNl(P ) ? UNl+1(P )j ? log jUNl?1(P ) ? UNl(P )j log 2 : The numerical results are presented in Table4 . They show that our quadrature method converges with order one even for solutions with low degree of smoothness.... ..."

### Table 1: Numerical experiments for the di usion equation with mixed boundary conditions.

"... In PAGE 14: ... The analytic solution is then known to be u(x; y) = sin(x) sin(y). The results of the experiments are given in the Table1 . As an outer iterative procedure, we use the Lanczos method of minimized iterations (see Section 3.... ..."

### Table 4 Pade (5; 6) Weighted Quadrature 3D Oscillatory Problem

"... In PAGE 11: ... The symmetric Lanczos algorithm is used to generate the Krylov subspaces. A subdiagonal Pad e (5; 6) approximation was chosen again as the exponential ap- proximation E(z) and numerical results for the weighted method (14) are summarized in Table4 . Test results for this problem are also reported in Table 6.... ..."

### Table 3 Pade (5; 6) Weighted Quadrature 3D unsymmetric Problem

"... In PAGE 10: ... When = 10 the discrete form of the di erential operator results in a nonsymmetric matrix and the Arnoldi algorithm must be employed to generate Krylov subspaces. Table3 contains a summary of the numerical results for the weighted quadrature method (14) applied to this problem. These results should be compared with Table 6.... In PAGE 11: ...Table 3 Pade (5; 6) Weighted Quadrature 3D unsymmetric Problem It is clear from Table3 that methods based on weighted quadrature can achieve a much smaller error (by ve orders of magnitude in some cases) for the same time step h and Krylov subspace dimension m used in the unweighted method (12). Furthermore, the number of quadrature points required to achieve this level of accuracy need not exceed npts = 3.... ..."