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Theory and Modern Applications

Table 5 The error norms and order of accuracy of the two schemes for Example 2 , 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

\(5.2002e^{-2}\)

\(4.8890e^{-2}\)

\(3.2290e^{-2}\)

\(7.1098e^{-3}\)

0.089

0.2

\(3.4421e^{-2}\)

\(2.2213e^{-2}\)

\(1.1180e^{-2}\)

\(5.3112e^{-3}\)

0.631

0.4

\(2.5541e^{-2}\)

\(1.7002e^{-2}\)

\(1.1040e^{-2}\)

\(3.1255e^{-4}\)

0.587

0.6

\(1.1983e^{-2}\)

\(7.2100e^{-3}\)

\(4.1033e^{-3}\)

\(6.1043e^{-5}\)

0.732

0.8

\(5.1287e^{-3}\)

\(1.9520e^{-3}\)

\(7.1530e^{-5}\)

\(1.6054e^{-5}\)

0.999

0.9

\(8.4161e^{-3}\)

\(4.2104e^{-3}\)

\(9.2610e^{-5}\)

\(2.6041e^{-5}\)

1.393

HOC

0.0

\(4.2122e^{-2}\)

\(3.3211e^{-3}\)

\(7.4412e^{-4}\)

\(5.8234e^{-4}\)

3.664

0.2

\(1.1218e^{-2}\)

\(1.6601e^{-3}\)

\(6.3318e^{-4}\)

\(4.7715e^{-5}\)

2.756

0.4

\(9.7110e^{-3}\)

\(6.1218e^{-4}\)

\(3.6152e^{-5}\)

\(3.1961e^{-6}\)

3.987

0.6

\(3.9328e^{-4}\)

\(4.1155e^{-5}\)

\(2.6001e^{-6}\)

\(2.3087e^{-6}\)

3.256

0.8

\(3.7022e^{-5}\)

\(1.8676e^{-6}\)

\(5.2260e^{-7}\)

\(3.3350e^{-7}\)

4.309

0.9

\(3.8844e^{-5}\)

\(2.8600e^{-6}\)

\(8.7210e^{-7}\)

\(5.3421e^{-7}\)

3.763