1 Introduction
2 Formulation of the continuous collocation polynomial approximation
3 Main theoretical results
3.1 Error estimates
3.2 Superconvergence of continuous trigonometric collocation polynomial approximations
3.3 Long-term behaviour of energy conservation
4 Numerical experiments
-
Trigonometric collocation polynomial approximation with Gauss nodes: By taking the nodes \(c_1,\cdots ,c_s\) as the zeros of the sth shifted Gauss-Legendre polynomialthen the Gauss quadrature formulas means that the integral (3.23) has order \(p=2s\). In this work, we choose the two-point Gauss-Legendre nodes$$\begin{aligned} \frac{\mathrm{d}^s}{\mathrm{d}x^s}\Big (x^s(x-1)^s\Big ), \end{aligned}$$and the three-point Gauss-Legendre nodes$$\begin{aligned} c_1=\frac{3-\sqrt{3}}{6},\qquad c_2=\frac{3+\sqrt{3}}{6}, \end{aligned}$$to construct the fourth-order and sixth-order trigonometric collocation time integrators, which are denoted by GTC2s4 and GTC3s6, respectively.$$\begin{aligned} c_1=\frac{5-\sqrt{15}}{10},\qquad c_2=\frac{1}{2},\qquad c_3=\frac{5+\sqrt{15}}{10}, \end{aligned}$$
-
Trigonometric collocation polynomial approximation with Lobatto nodes: If we take the nodes \(c_1,\cdots ,c_s\) as the zeros of the sth Legendre polynomialthen the Lobatto quadrature formulas lead the integral (3.23) could achieve the highest possible order \(p=2s-2\). Similarly, by taking the three-point Lobatto nodes$$\begin{aligned} \frac{\mathrm{d}^{s-2}}{\mathrm{d}x^{s-2}}\Big (x^{s-1}(x-1)^{s-1}\Big ), \end{aligned}$$and the four-point Lobatto nodes$$\begin{aligned} c_1=0,\qquad c_2=\frac{1}{2},\qquad c_3=1, \end{aligned}$$we can derive the fourth-order and sixth-order trigonometric collocation time integrators, and denoted as LTC2s4 and LTC3s6, respectively.$$\begin{aligned} c_1=0,\qquad c_2=\frac{5-\sqrt{5}}{10},\qquad c_3=\frac{5+\sqrt{5}}{10},\qquad c_4=1, \end{aligned}$$
-
BH1: the symmetric Birkhoff-Hermite time integrator of order four derived in [22]
-
BH2: the symmetric Birkhoff-Hermite time integrator of order six derived in [22]
-
GAS2s4: the two-stage Gauss time integration method of order four presented in [16];
-
GAS3s6: the three-stage Gauss time integration method of order six presented in [16];
-
LIIIA3s4: the Labatto IIIA method of order four presented in [16];
-
LIIIA4s6: the Labatto IIIA method of order six presented in [16].
-
ERKN3s4: the three-stage explicit ERKN method of order four derived in [38];
-
ERKN7s6: the seven-stage explicit ERKN method of order six presented in [33];
h | GTC2s4 | GTC3s6 | LTC3s4 | LTC4s6 | ||||
---|---|---|---|---|---|---|---|---|
Error | Rate | Error | Rate | Error | Rate | Error | Rate | |
h | \(6.7910\mathrm{E}-05\) | * | \(4.5151\mathrm{E}-07\) | * | \(7.1473\mathrm{E}-05\) | * | \(5.2011\mathrm{E}-07\) | * |
h/2 | \(4.0054\mathrm{E}-06\) | 4.0836 | \(5.9649\mathrm{E}-09\) | 6.2421 | \(4.1139\mathrm{E}-06\) | 4.1188 | \(7.1630\mathrm{E}-09\) | 6.1821 |
\(h/2^2\) | \(2.4725\mathrm{E}-07\) | 4.0179 | \(8.9788\mathrm{E}-11\) | 6.0538 | \(2.5238\mathrm{E}-07\) | 4.0268 | \(1.0944\mathrm{E}-10\) | 6.0324 |
\(h/2^3\) | \(1.5407\mathrm{E}-08\) | 4.0043 | \(1.4045\mathrm{E}-12\) | 5.9984 | \(1.5702\mathrm{E}-08\) | 4.0066 | \(1.7022\mathrm{E}-12\) | 6.0065 |
h
| GTC2s4 | GTC3s6 | LTC3s4 | LTC4s6 | ||||
---|---|---|---|---|---|---|---|---|
Error | Rate | Error | Rate | Error | Rate | Error | Rate | |
h
|
\(4.3034\mathrm{E}-04\)
| * |
\(4.4094\mathrm{E}-06\)
| * |
\(2.9664\mathrm{E}-04\)
| * |
\(9.2841\mathrm{E}-06\)
| * |
h/2 |
\(2.3794\mathrm{E}-05\)
| 4.1768 |
\(1.0708\mathrm{E}-08\)
| 8.6857 |
\(2.3478\mathrm{E}-05\)
| 3.6593 |
\(9.1543\mathrm{E}-08\)
| 6.6642 |
\(h/2^2\)
|
\(1.4706\mathrm{E}-06\)
| 4.0161 |
\(1.9491\mathrm{E}-10\)
| 5.7798 |
\(1.5089\mathrm{E}-06\)
| 3.9597 |
\(1.3729\mathrm{E}-09\)
| 6.0592 |
\(h/2^3\)
|
\(9.1694\mathrm{E}-08\)
| 4.0034 |
\(3.2456\mathrm{E}-12\)
| 5.9082 |
\(9.4910\mathrm{E}-08\)
| 3.9908 |
\(2.1330\mathrm{E}-11\)
| 6.0081 |
h
| GTC2s4 | GTC3s6 | LTC3s4 | LTC4s6 | ||||
---|---|---|---|---|---|---|---|---|
Error | Rate | Error | Rate | Error | Rate | Error | Rate | |
h
|
\(9.7402\mathrm{E}-04\)
| * |
\(8.8434\mathrm{E}-06\)
| * |
\(1.3353\mathrm{E}-03\)
| * |
\(9.1889\mathrm{E}-06\)
| * |
h/2 |
\(6.3235\mathrm{E}-05\)
| 3.9452 |
\(1.3105\mathrm{E}-07\)
| 6.0764 |
\(8.1005\mathrm{E}-05\)
| 4.0430 |
\(1.2996\mathrm{E}-07\)
| 6.1437 |
\(h/2^2\)
|
\(3.8634\mathrm{E}-06\)
| 4.0328 |
\(1.9952\mathrm{E}-09\)
| 6.0375 |
\(5.0517\mathrm{E}-06\)
| 4.0032 |
\(2.0248\mathrm{E}-09\)
| 6.0042 |
\(h/2^3\)
|
\(2.4042\mathrm{E}-07\)
| 4.0062 |
\(3.1321\mathrm{E}-11\)
| 5.9933 |
\(3.1464\mathrm{E}-07\)
| 4.0050 |
\(3.1393\mathrm{E}-11\)
| 6.0112 |
h
| GTC2s4 | GTC3s6 | LTC3s4 | LTC4s6 | ||||
---|---|---|---|---|---|---|---|---|
Error | Rate | Error | Rate | Error | Rate | Error | Rate | |
h
|
\(5.1140\mathrm{E}-02\)
| * |
\(2.2981\mathrm{E}-03\)
| * |
\(6.3274\mathrm{E}-02\)
| * |
\(2.4464\mathrm{E}-03\)
| * |
h/2 |
\(1.7342\mathrm{E}-03\)
| 4.8822 |
\(1.6195\mathrm{E}-05\)
| 7.1487 |
\(2.3705\mathrm{E}-03\)
| 4.7384 |
\(1.6012\mathrm{E}-05\)
| 7.2554 |
\(h/2^2\)
|
\(9.8471\mathrm{E}-05\)
| 4.1384 |
\(2.2820\mathrm{E}-07\)
| 6.1491 |
\(1.3679\mathrm{E}-04\)
| 4.1152 |
\(2.2322\mathrm{E}-07\)
| 6.1645 |
\(h/2^3\)
|
\(6.0027\mathrm{E}-06\)
| 4.0360 |
\(3.4719\mathrm{E}-09\)
| 6.0385 |
\(8.3830\mathrm{E}-06\)
| 4.0284 |
\(3.3827\mathrm{E}-09\)
| 6.0442 |
h | GTC2s4 | GTC3s6 | LTC3s4 | LTC4s6 | ||||
---|---|---|---|---|---|---|---|---|
Error | Rate | Error | Rate | Error | Rate | Error | Rate | |
h | \(8.2434\mathrm{E}-05\) | * | \(2.6349\mathrm{E}-06\) | * | \(1.2226\mathrm{E}-04\) | * | \(3.4091\mathrm{E}-06\) | * |
h/2 | \(4.9537\mathrm{E}-06\) | 4.0567 | \(4.9539\mathrm{E}-08\) | 5.7331 | \(7.5801\mathrm{E}-06\) | 4.0116 | \(6.5801\mathrm{E}-08\) | 5.6951 |
\(h/2^2\) | \(3.0966\mathrm{E}-07\) | 3.9997 | \(8.1146\mathrm{E}-10\) | 5.9319 | \(4.7386\mathrm{E}-07\) | 3.9997 | \(1.0826\mathrm{E}-09\) | 5.9255 |
\(h/2^3\) | \(1.9359\mathrm{E}-08\) | 3.9996 | \(1.2790\mathrm{E}-11\) | 5.9875 | \(2.9621\mathrm{E}-08\) | 3.9997 | \(1.7079\mathrm{E}-11\) | 5.9862 |
h
| GTC2s4 | GTC3s6 | LTC3s4 | LTC4s6 | ||||
---|---|---|---|---|---|---|---|---|
Error | Rate | Error | Rate | Error | Rate | Error | Rate | |
h
|
\(1.8869\mathrm{E}-06\)
| * |
\(6.3650\mathrm{E}-09\)
| * |
\(3.1193\mathrm{E}-06\)
| * |
\(9.0491\mathrm{E}-09\)
| * |
h/2 |
\(1.1854\mathrm{E}-07\)
| 3.9926 |
\(9.1859\mathrm{E}-11\)
| 6.1146 |
\(1.9403\mathrm{E}-07\)
| 4.0069 |
\(1.3681\mathrm{E}-10\)
| 6.0475 |
\(h/2^2\)
|
\(7.5259\mathrm{E}-09\)
| 3.9773 |
\(1.4297\mathrm{E}-12\)
| 6.0056 |
\(1.2354\mathrm{E}-08\)
| 3.9733 |
\(2.1194\mathrm{E}-12\)
| 6.0124 |
\(h/2^3\)
|
\(4.6940\mathrm{E}-10\)
| 4.0030 |
\(4.2044\mathrm{E}-13\)
| – |
\(7.7083\mathrm{E}-10\)
| 4.0024 |
\(4.1939\mathrm{E}-13\)
| – |
h | GTC2s4 | GTC3s6 | LTC3s4 | LTC4s6 | ||||
---|---|---|---|---|---|---|---|---|
Error | Rate | Error | Rate | Error | Rate | Error | Rate | |
h | \(2.2948\mathrm{E}-04\) | * | \(6.5535\mathrm{E}-06\) | * | \(3.3743\mathrm{E}-04\) | * | \(8.7509\mathrm{E}-06\) | * |
h/2 | \(1.5263\mathrm{E}-05\) | 3.9102 | \(1.0957\mathrm{E}-07\) | 5.9024 | \(2.2811\mathrm{E}-05\) | 3.8868 | \(1.4485\mathrm{E}-07\) | 5.9168 |
\(h/2^2\) | \(9.6938\mathrm{E}-07\) | 3.9768 | \(1.7381\mathrm{E}-09\) | 5.9782 | \(1.4532\mathrm{E}-06\) | 3.9724 | \(2.3046\mathrm{E}-09\) | 5.9739 |
\(h/2^3\) | \(6.0899\mathrm{E}-08\) | 3.9926 | \(2.8857\mathrm{E}-11\) | 5.9124 | \(9.1311\mathrm{E}-08\) | 3.9923 | \(3.7772\mathrm{E}-11\) | 5.9311 |
h
| GTC2s4 | GTC3s6 | LTC3s4 | LTC4s6 | ||||
---|---|---|---|---|---|---|---|---|
Error | Rate | Error | Rate | Error | Rate | Error | Rate | |
h
|
\(1.1468\mathrm{E}-04\)
| * |
\(3.2996\mathrm{E}-06\)
| * |
\(1.6896\mathrm{E}-04\)
| * |
\(4.3554\mathrm{E}-06\)
| * |
h/2 |
\(7.6411\mathrm{E}-06\)
| 3.9077 |
\(5.4632\mathrm{E}-08\)
| 5.9164 |
\(1.1406\mathrm{E}-05\)
| 3.8888 |
\(7.2744\mathrm{E}-08\)
| 5.9038 |
\(h/2^2\)
|
\(4.8518\mathrm{E}-07\)
| 3.9772 |
\(8.6855\mathrm{E}-10\)
| 5.9750 |
\(7.2682\mathrm{E}-07\)
| 3.9721 |
\(1.1541\mathrm{E}-09\)
| 5.9779 |
\(h/2^3\)
|
\(3.0467\mathrm{E}-08\)
| 3.9932 |
\(1.5864\mathrm{E}-11\)
| 5.7748 |
\(4.5693\mathrm{E}-08\)
| 3.9916 |
\(2.0141\mathrm{E}-11\)
| 5.8405 |
h
| GTC2s4 | GTC3s6 | LTC3s4 | LTC4s6 | ||||
---|---|---|---|---|---|---|---|---|
Error | Rate | Error | Rate | Error | Rate | Error | Rate | |
h
|
\(2.6403\mathrm{E}-04\)
| * |
\(4.5557\mathrm{E}-05\)
| * |
\(3.2219\mathrm{E}-04\)
| * |
\(4.8043\mathrm{E}-05\)
| * |
h/2 | 1.2309E − 05 | 4.4229 |
\(4.2258\mathrm{E}-07\)
| 6.7523 |
\(1.2481\mathrm{E}-05\)
| 4.6901 |
\(1.2309\mathrm{E}-05\)
| 6.5086 |
\(h/2^2\)
| 4.7861E − 07 | 4.6847 |
\(3.9796\mathrm{E}-09\)
| 6.7305 |
\(4.5088\mathrm{E}-07\)
| 4.7909 |
\(4.7861\mathrm{E}-07\)
| 6.6884 |
\(h/2^3\)
| 2.7186E − 08 | 4.1379 |
\(2.3868\mathrm{E}-10\)
| 4.0595 |
\(2.5265\mathrm{E}-08\)
| 4.1575 |
\(2.7186\mathrm{E}-08\)
| 4.7938 |
h
| GTC2s4 | GTC3s6 | LTC3s4 | LTC4s6 | ||||
---|---|---|---|---|---|---|---|---|
Error | Rate | Error | Rate | Error | Rate | Error | Rate | |
h
|
\(2.6403\mathrm{E}-04\)
| * |
\(4.5556\mathrm{E}-05\)
| * |
\(3.2219\mathrm{E}-04\)
| * |
\(4.8043\mathrm{E}-05\)
| * |
h/2 |
\(1.2309\mathrm{E}-05\)
| 4.4229 |
\(4.2276\mathrm{E}-07\)
| 6.7517 |
\(1.2481\mathrm{E}-05\)
| 4.6901 |
\(5.2763\mathrm{E}-07\)
| 6.5087 |
\(h/2^2\)
|
\(4.7861\mathrm{E}-07\)
| 4.6847 |
\(4.6871\mathrm{E}-09\)
| 6.4950 |
\(4.5159\mathrm{E}-07\)
| 4.7886 |
\(5.1139\mathrm{E}-09\)
| 6.6889 |
\(h/2^3\)
|
\(2.7185\mathrm{E}-08\)
| 4.1380 |
\(1.5256\mathrm{E}-09\)
| – |
\(2.6544\mathrm{E}-08\)
| 4.0886 |
\(1.4599\mathrm{E}-09\)
| – |