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Theory and Modern Applications

Table 3 The error norms and order of accuracy of the two schemes for Example 1 , where \(\pmb{e^{-5}=10^{-5}}\) , \(\pmb{\|e\|_{2}}\) , \(\pmb{k = 10}\) , \(\pmb{N = 16,32,64,128}\)

From: Multigrid method based on transformation-free high-order scheme for solving 2D Helmholtz equation on nonuniform grids

N

λ

\(\boldsymbol{16^{2}} \)

\(\boldsymbol{32^{2}}\)

\(\boldsymbol{64^{2}}\)

\(\boldsymbol{128^{2}}\)

Order

CDS

0.0

\(6.0982e^{-4}\)

\(4.9180e^{-4}\)

\(9.9082e^{-5}\)

\(7.9118e^{-5}\)

0.310

−0.2

\(4.1044e^{-4}\)

\(3.1322e^{-4}\)

\(7.1160e^{-5}\)

\(5.0032e^{-5}\)

0.392

−0.4

\(2.5100e^{-4}\)

\(1.7200e^{-4}\)

\(6.7640e^{-5}\)

\(3.9155e^{-5}\)

0.545

−0.6

\(8.1398e^{-5}\)

\(6 .1021e^{-5}\)

\(4.3302e^{-5}\)

\(2.4203e^{-5}\)

0.415

−0.8

\(5.8197e^{-5}\)

\(5.0295e^{-5}\)

\(3.0955e^{-5}\)

\(1.5064e^{-5}\)

0.531

−0.9

\(3.4623e^{-5}\)

\(2.4504e^{-5}\)

\(2.1089e^{-5}\)

\(3.2161e^{-5}\)

0.210

HOC

0.0

\(4.8122e^{-4}\)

\(3.6113e^{-4}\)

\(9.4102e^{-5}\)

\(5.8410e^{-5}\)

0.414

−0.2

\(1.1438e^{-4}\)

\(1.0661e^{-4}\)

\(6.3918e^{-5}\)

\(4.1235e^{-5}\)

0.101

−0.4

\(9.1100e^{-5}\)

\(6.1818e^{-5}\)

\(3.2561e^{-5}\)

\(1.9760e^{-5}\)

0.559

−0.6

\(7.1398e^{-5}\)

\(4.5502e^{-5}\)

\(1.6021e^{-5}\)

\(6.2030e^{-6}\)

0.650

−0.8

\(4.7341e^{-5}\)

\(8.8061e^{-6}\)

\(7.6660e^{-6}\)

\(3.7210e^{-7}\)

0.242

−0.9

\(7.2104e^{-5}\)

\(1.1268e^{-5}\)

\(9.9271e^{-6}\)

\(9.3180e^{-7}\)

2.677