In this article, we study a 2D nonlinear time-fractional Rayleigh-Stokes problem, which has an anomalous sub-diffusion term, on triangular meshes by quadratic finite volume element schemes. Time-fractional derivative, defined by Caputo fractional derivative, is discretized through
Recently, due to the widespread use of fractional partial differential equations (FPDEs), such as dispersion in a porous medium, statistical mechanics, mathematical biology and so on, numerical solution of FPDEs becomes one of the frontier fields in the research. Fractional partial differential equations can be roughly classified into three categories: Space FPDEs [
At present, numerical simulation is an important and effective way to solve partial differential equations, and the relevant numerical methods can be finite difference methods [
There are some research about quadratic finite volume element methods for solving partial differential equations on triangular meshes. Tian et al. [
In this article, the quadratic finite volume element method is proposed to solve one class of FPDEs, that is, a 2D nonlinear time-fractional Rayleigh-Stokes problem with the time-fractional derivative defined by Caputo fractional derivative. In spatial direction, this problem is solved by a class of quadratic finite volume element schemes with two parameters
The outline of this paper is as follows. In Section 2, we describe in details the specific algorithm steps of the quadratic finite volume element schemes over triangular meshes, and finally obtain the fully discrete schemes. In Section 3, some numerical experiments are performed to investigate the performance of the quadratic finite volume element schemes. The numerical results are also compared with those of an existing quadratic finite element scheme. A brief conclusion ends this article in last section.
In this article, we construct a family of quadratic finite volume element schemes to solve the following 2D nonlinear time-fractional Rayleigh-Stokes problem:
where
For the numerical solution of
In order to define the test function space, we need to construct a dual mesh associated with
while
where
In this part, we propose the following spatial discrete formulation of
By the divergence theorem and the definition of
where
For the computation of the terms in
where
where
Next we introduce the fully discrete schemes at time
where
Based on Lemmas 2.1–2.3, we propose the following fully discrete schemes by the
for
for
where
Let the basis functions of the trial function space
for
for
where
mass matrix
where
For the above finite volume element schemes, we emphasize that the nonlinear term is approximated by using the linearized difference scheme in Lemma 2.2, and none nonlinear iteration is involved. The whole algorithm is summarized below.
In this section, we use First scheme (QFVE-1): Second scheme (QFVE-2): Third scheme (QFVE-3): Fourth scheme (QFVE-4):
We remark that the counterparts of the above schemes for elliptic problems have been studied in [
Solve
Scheme | |||||||
---|---|---|---|---|---|---|---|
QFE | 0.1 | 1.5374E-02 | 1.9390E-03 | 2.4358E-04 | 3.0502E-05 | 3.8152E-06 | |
2.98711 | 2.99288 | 2.99738 | 2.99909 | ||||
0.2 | 1.5378E-02 | 1.9392E-03 | 2.4359E-04 | 3.0504E-05 | 3.8157E-06 | ||
2.98730 | 2.99296 | 2.99738 | 2.99897 | ||||
0.5 | 1.5394E-02 | 1.9400E-03 | 2.4362E-04 | 3.0507E-05 | 3.8170E-06 | ||
2.98824 | 2.99335 | 2.99741 | 2.99862 | ||||
0.9 | 1.5426E-02 | 1.9415E-03 | 2.4368E-04 | 3.0512E-05 | 3.8190E-06 | ||
2.99010 | 2.99413 | 2.99753 | 2.99810 | ||||
QFVE-1 | 0.1 | 1.4782E-02 | 1.9197E-03 | 2.4302E-04 | 3.0486E-05 | 3.8147E-06 | |
2.94484 | 2.98177 | 2.99483 | 2.99850 | ||||
0.2 | 1.4783E-02 | 1.9198E-03 | 2.4302E-04 | 3.0487E-05 | 3.8150E-06 | ||
2.94493 | 2.98180 | 2.99482 | 2.99841 | ||||
0.5 | 1.4790E-02 | 1.9201E-03 | 2.4303E-04 | 3.0489E-05 | 3.8160E-06 | ||
2.94537 | 2.98196 | 2.99480 | 2.99815 | ||||
0.9 | 1.4805E-02 | 1.9208E-03 | 2.4306E-04 | 3.0492E-05 | 3.8175E-06 | ||
2.94633 | 2.98230 | 2.99483 | 2.99772 | ||||
QFVE-2 | 0.1 | 1.6242E-02 | 2.2118E-03 | 3.1919E-04 | 5.5340E-05 | 1.1798E-05 | |
2.87640 | 2.79275 | 2.52803 | 2.22973 | ||||
0.2 | 1.6244E-02 | 2.2123E-03 | 3.1933E-04 | 5.5389E-05 | 1.1819E-05 | ||
2.87633 | 2.79243 | 2.52736 | 2.22850 | ||||
0.5 | 1.6257E-02 | 2.2142E-03 | 3.1986E-04 | 5.5573E-05 | 1.1889E-05 | ||
2.87615 | 2.79128 | 2.52500 | 2.22480 | ||||
0.9 | 1.6281E-02 | 2.2177E-03 | 3.2077E-04 | 5.5874E-05 | 1.1996E-05 | ||
2.87601 | 2.78944 | 2.52131 | 2.21962 | ||||
QFVE-3 | 0.1 | 1.4931E-02 | 4.5874E-03 | 1.2418E-03 | 3.1723E-04 | 7.9750E-05 | |
1.70251 | 1.88525 | 1.96884 | 1.99197 | ||||
0.2 | 1.4945E-02 | 4.5932E-03 | 1.2434E-03 | 3.1762E-04 | 7.9841E-05 | ||
1.70211 | 1.88524 | 1.96887 | 1.99209 | ||||
0.5 | 1.4998E-02 | 4.6150E-03 | 1.2494E-03 | 3.1914E-04 | 8.0206E-05 | ||
1.70039 | 1.88511 | 1.96894 | 1.99242 | ||||
0.9 | 1.5081E-02 | 4.6506E-03 | 1.2593E-03 | 3.2168E-04 | 8.0821E-05 | ||
1.69725 | 1.88475 | 1.96897 | 1.99283 | ||||
QFVE-4 | 0.1 | 3.2294E-02 | 9.3096E-03 | 2.4168E-03 | 6.0999E-04 | 1.5287E-04 | |
1.79448 | 1.94560 | 1.98626 | 1.99652 | ||||
0.2 | 3.2345E-02 | 9.3225E-03 | 2.4200E-03 | 6.1079E-04 | 1.5306E-04 | ||
1.79477 | 1.94567 | 1.98629 | 1.99659 | ||||
0.5 | 3.2537E-02 | 9.3717E-03 | 2.4324E-03 | 6.1387E-04 | 1.5381E-04 | ||
1.79568 | 1.94592 | 1.98639 | 1.99678 | ||||
0.9 | 3.2842E-02 | 9.4524E-03 | 2.4529E-03 | 6.1897E-04 | 1.5506E-04 | ||
1.79681 | 1.94622 | 1.98652 | 1.99701 |
Scheme | |||||||
---|---|---|---|---|---|---|---|
QFE | 0.1 | 1.6991E-02 | 2.1497E-03 | 2.6744E-04 | 3.3367E-05 | 4.1695E-06 | |
2.98255 | 3.00683 | 3.00271 | 3.00050 | ||||
0.2 | 1.6996E-02 | 2.1499E-03 | 2.6745E-04 | 3.3369E-05 | 4.1700E-06 | ||
2.98284 | 3.00691 | 3.00270 | 3.00039 | ||||
0.5 | 1.7016E-02 | 2.1507E-03 | 2.6748E-04 | 3.3372E-05 | 4.1713E-06 | ||
2.98402 | 3.00728 | 3.00274 | 3.00007 | ||||
0.9 | 1.7054E-02 | 2.1524E-03 | 2.6755E-04 | 3.3377E-05 | 4.1733E-06 | ||
2.98605 | 3.00806 | 3.00288 | 2.99959 | ||||
QFVE-1 | 0.1 | 1.6436E-02 | 2.1287E-03 | 2.6697E-04 | 3.3367E-05 | 4.1703E-06 | |
2.94882 | 2.99519 | 3.00019 | 3.00021 | ||||
0.2 | 1.6439E-02 | 2.1288E-03 | 2.6698E-04 | 3.3368E-05 | 4.1706E-06 | ||
2.94907 | 2.99524 | 3.00018 | 3.00012 | ||||
0.5 | 1.6454E-02 | 2.1292E-03 | 2.6699E-04 | 3.3370E-05 | 4.1716E-06 | ||
2.95000 | 2.99544 | 3.00018 | 2.99988 | ||||
0.9 | 1.6477E-02 | 2.1301E-03 | 2.6703E-04 | 3.3373E-05 | 4.1732E-06 | ||
2.95148 | 2.99585 | 3.00023 | 2.99948 | ||||
QFVE-2 | 0.1 | 1.7951E-02 | 2.4431E-03 | 3.4969E-04 | 6.0279E-05 | 1.2808E-05 | |
2.87733 | 2.80453 | 2.53636 | 2.23464 | ||||
0.2 | 1.7957E-02 | 2.4438E-03 | 3.4989E-04 | 6.0346E-05 | 1.2833E-05 | ||
2.87741 | 2.80415 | 2.53556 | 2.23336 | ||||
0.5 | 1.7981E-02 | 2.4465E-03 | 3.5063E-04 | 6.0592E-05 | 1.2921E-05 | ||
2.87764 | 2.80273 | 2.53273 | 2.22942 | ||||
0.9 | 1.8019E-02 | 2.4513E-03 | 3.5186E-04 | 6.0990E-05 | 1.3055E-05 | ||
2.87792 | 2.80051 | 2.52835 | 2.22391 | ||||
QFVE-3 | 0.1 | 1.6936E-02 | 4.9637E-03 | 1.3456E-03 | 3.4475E-04 | 8.6783E-05 | |
1.77061 | 1.88317 | 1.96464 | 1.99006 | ||||
0.2 | 1.6954E-02 | 4.9713E-03 | 1.3477E-03 | 3.4527E-04 | 8.6907E-05 | ||
1.76990 | 1.88316 | 1.96467 | 1.99018 | ||||
0.5 | 1.7019E-02 | 4.9998E-03 | 1.3555E-03 | 3.4727E-04 | 8.7391E-05 | ||
1.76720 | 1.88302 | 1.96475 | 1.99049 | ||||
0.9 | 1.7124E-02 | 5.0457E-03 | 1.3683E-03 | 3.5053E-04 | 8.8189E-05 | ||
1.76284 | 1.88265 | 1.96481 | 1.99086 | ||||
QFVE-4 | 0.1 | 3.5054E-02 | 1.0067E-02 | 2.6271E-03 | 6.6460E-04 | 1.6671E-04 | |
1.79998 | 1.93807 | 1.98289 | 1.99513 | ||||
0.2 | 3.5115E-02 | 1.0083E-02 | 2.6312E-03 | 6.6563E-04 | 1.6696E-04 | ||
1.80014 | 1.93816 | 1.98292 | 1.99519 | ||||
0.5 | 3.5345E-02 | 1.0145E-02 | 2.6469E-03 | 6.6956E-04 | 1.6793E-04 | ||
1.80067 | 1.93845 | 1.98303 | 1.99537 | ||||
0.9 | 3.5715E-02 | 1.0247E-02 | 2.6726E-03 | 6.7598E-04 | 1.6951E-04 | ||
1.80139 | 1.93883 | 1.98318 | 1.99558 |
Scheme | ||||||
---|---|---|---|---|---|---|
QFE | 0.1 | 3.7630E-03 | 1.4679E-03 | 4.6863E-04 | 1.3264E-04 | |
1.35809 | 1.64726 | 1.82090 | ||||
0.2 | 5.2654E-03 | 1.6715E-03 | 4.9435E-04 | 1.3573E-04 | ||
1.65542 | 1.75752 | 1.86482 | ||||
0.5 | 1.2032E-02 | 3.0704E-03 | 7.7387E-04 | 1.9313E-04 | ||
1.97040 | 1.98825 | 2.00251 | ||||
0.9 | 2.1267E-02 | 5.3181E-03 | 1.3141E-03 | 3.2290E-04 | ||
1.99964 | 2.01682 | 2.02493 | ||||
QFVE-1 | 0.1 | 3.7630E-03 | 1.4679E-03 | 4.6863E-04 | 1.3264E-04 | |
1.35809 | 1.64725 | 1.82091 | ||||
0.2 | 5.2654E-03 | 1.6715E-03 | 4.9435E-04 | 1.3573E-04 | ||
1.65542 | 1.75752 | 1.86483 | ||||
0.5 | 1.2032E-02 | 3.0704E-03 | 7.7387E-04 | 1.9313E-04 | ||
1.97040 | 1.98825 | 2.00253 | ||||
0.9 | 2.1267E-02 | 5.3181E-03 | 1.3141E-03 | 3.2290E-04 | ||
1.99964 | 2.01683 | 2.02494 | ||||
QFVE-2 | 0.1 | 3.7637E-03 | 1.4681E-03 | 4.6863E-04 | 1.3258E-04 | |
1.35817 | 1.64744 | 1.82161 | ||||
0.2 | 5.2671E-03 | 1.6726E-03 | 4.9519E-04 | 1.3642E-04 | ||
1.65489 | 1.75607 | 1.85989 | ||||
0.5 | 1.2035E-02 | 3.0756E-03 | 7.7601E-04 | 1.9518E-04 | ||
1.96962 | 1.98534 | 1.99130 | ||||
0.9 | 2.1269E-02 | 5.3206E-03 | 1.3166E-03 | 3.2538E-04 | ||
1.99913 | 2.01477 | 2.01663 | ||||
QFVE-3 | 0.1 | 3.7592E-03 | 1.4679E-03 | 4.7026E-04 | 1.3592E-04 | |
1.35667 | 1.64220 | 1.79072 | ||||
0.2 | 5.2539E-03 | 1.6643E-03 | 4.8986E-04 | 1.3350E-04 | ||
1.65845 | 1.76449 | 1.87549 | ||||
0.5 | 1.2016E-02 | 3.0549E-03 | 7.5950E-04 | 1.7996E-04 | ||
1.97572 | 2.00801 | 2.07741 | ||||
0.9 | 2.1250E-02 | 5.3007E-03 | 1.2968E-03 | 3.0587E-04 | ||
2.00320 | 2.03119 | 2.08398 | ||||
QFVE-4 | 0.1 | 3.7566E-03 | 1.4689E-03 | 4.7334E-04 | 1.4214E-04 | |
1.35466 | 1.63384 | 1.73558 | ||||
0.2 | 5.2442E-03 | 1.6589E-03 | 4.8737E-04 | 1.3499E-04 | ||
1.66054 | 1.76712 | 1.85212 | ||||
0.5 | 1.2001E-02 | 3.0415E-03 | 7.4735E-04 | 1.6998E-04 | ||
1.98033 | 2.02493 | 2.13638 | ||||
0.9 | 2.1235E-02 | 5.2854E-03 | 1.2817E-03 | 2.9133E-04 | ||
2.00635 | 2.04395 | 2.13736 |
Scheme | ||||||
---|---|---|---|---|---|---|
QFE | 0.1 | 3.7630E-03 | 1.4679E-03 | 4.6863E-04 | 1.3264E-04 | |
1.35809 | 1.64726 | 1.82090 | ||||
0.2 | 5.2654E-03 | 1.6715E-03 | 4.9435E-04 | 1.3573E-04 | ||
1.65541 | 1.75752 | 1.86482 | ||||
0.5 | 1.2032E-02 | 3.0704E-03 | 7.7388E-04 | 1.9313E-04 | ||
1.97040 | 1.98825 | 2.00250 | ||||
0.9 | 2.1267E-02 | 5.3181E-03 | 1.3141E-03 | 3.2290E-04 | ||
1.99964 | 2.01682 | 2.02492 | ||||
QFVE-1 | 0.1 | 3.7630E-03 | 1.4679E-03 | 4.6863E-04 | 1.3264E-04 | |
1.35809 | 1.64726 | 1.82091 | ||||
0.2 | 5.2654E-03 | 1.6715E-03 | 4.9435E-04 | 1.3573E-04 | ||
1.65542 | 1.75752 | 1.86483 | ||||
0.5 | 1.2032E-02 | 3.0704E-03 | 7.7387E-04 | 1.9313E-04 | ||
1.97040 | 1.98825 | 2.00252 | ||||
0.9 | 2.1267E-02 | 5.3181E-03 | 1.3141E-03 | 3.2290E-04 | ||
1.99964 | 2.01683 | 2.02493 | ||||
QFVE-2 | 0.1 | 3.7634E-03 | 1.4678E-03 | 4.6827E-04 | 1.3222E-04 | |
1.35839 | 1.64821 | 1.82445 | ||||
0.2 | 5.2670E-03 | 1.6724E-03 | 4.9492E-04 | 1.3614E-04 | ||
1.65505 | 1.75667 | 1.86215 | ||||
0.5 | 1.2035E-02 | 3.0726E-03 | 7.7596E-04 | 1.9511E-04 | ||
1.96963 | 1.98542 | 1.99170 | ||||
0.9 | 2.1270E-02 | 5.3207E-03 | 1.3167E-03 | 3.2544E-04 | ||
1.99911 | 2.01472 | 2.01643 | ||||
QFVE-3 | 0.1 | 3.7608E-03 | 1.4700E-03 | 4.7249E-04 | 1.3833E-04 | |
1.35529 | 1.63741 | 1.77221 | ||||
0.2 | 5.2545E-03 | 1.6656E-03 | 4.9154E-04 | 1.3552E-04 | ||
1.65748 | 1.76069 | 1.85881 | ||||
0.5 | 1.2015E-02 | 3.0548E-03 | 7.5967E-04 | 1.8046E-04 | ||
1.97572 | 2.00764 | 2.07370 | ||||
0.9 | 2.1249E-02 | 5.3001E-03 | 1.2963E-03 | 3.0543E-04 | ||
2.00332 | 2.03165 | 2.08546 | ||||
QFVE-4 | 0.1 | 3.7596E-03 | 1.4726E-03 | 4.7739E-04 | 1.4668E-04 | |
1.35219 | 1.62516 | 1.70246 | ||||
0.2 | 5.2453E-03 | 1.6612E-03 | 4.9042E-04 | 1.3897E-04 | ||
1.65882 | 1.76009 | 1.81921 | ||||
0.5 | 1.2000E-02 | 3.0412E-03 | 7.4760E-04 | 1.7120E-04 | ||
1.98038 | 2.02429 | 2.12658 | ||||
0.9 | 2.1233E-02 | 5.2841E-03 | 1.2806E-03 | 2.9057E-04 | ||
2.00660 | 2.04486 | 2.13982 |
Solve
Scheme | |||||||
---|---|---|---|---|---|---|---|
QFE | 0.1 | 3.4789E-02 | 4.4415E-03 | 5.3823E-04 | 6.6439E-05 | 8.2777E-06 | |
2.96951 | 3.04474 | 3.01812 | 3.00474 | ||||
0.2 | 3.4787E-02 | 4.4414E-03 | 5.3823E-04 | 6.6440E-05 | 8.2779E-06 | ||
2.96947 | 3.04472 | 3.01811 | 3.00471 | ||||
0.5 | 3.4783E-02 | 4.4412E-03 | 5.3822E-04 | 6.6440E-05 | 8.2784E-06 | ||
2.96936 | 3.04466 | 3.01807 | 3.00463 | ||||
0.9 | 3.4778E-02 | 4.4409E-03 | 5.3821E-04 | 6.6441E-05 | 8.2790E-06 | ||
2.96924 | 3.04460 | 3.01803 | 3.00455 | ||||
QFVE-1 | 0.1 | 3.1871E-02 | 4.2336E-03 | 5.2968E-04 | 6.6155E-05 | 8.2687E-06 | |
2.91229 | 2.99871 | 3.00120 | 3.00011 | ||||
0.2 | 3.1870E-02 | 4.2336E-03 | 5.2968E-04 | 6.6155E-05 | 8.2689E-06 | ||
2.91226 | 2.99870 | 3.00119 | 3.00008 | ||||
0.5 | 3.1867E-02 | 4.2335E-03 | 5.2967E-04 | 6.6156E-05 | 8.2693E-06 | ||
2.91217 | 2.99866 | 3.00116 | 3.00002 | ||||
0.9 | 3.1864E-02 | 4.2333E-03 | 5.2967E-04 | 6.6156E-05 | 8.2698E-06 | ||
2.91208 | 2.99862 | 3.00114 | 2.99996 | ||||
QFVE-2 | 0.1 | 3.2196E-02 | 4.4497E-03 | 6.0169E-04 | 9.2053E-05 | 1.7519E-05 | |
2.85511 | 2.88661 | 2.70849 | 2.39355 | ||||
0.2 | 3.2195E-02 | 4.4495E-03 | 6.0166E-04 | 9.2044E-05 | 1.7518E-05 | ||
2.85510 | 2.88664 | 2.70855 | 2.39348 | ||||
0.5 | 3.2192E-02 | 4.4492E-03 | 6.0158E-04 | 9.2019E-05 | 1.7515E-05 | ||
2.85507 | 2.88672 | 2.70874 | 2.39335 | ||||
0.9 | 3.2188E-02 | 4.4488E-03 | 6.0151E-04 | 9.2003E-05 | 1.7515E-05 | ||
2.85503 | 2.88676 | 2.70884 | 2.39313 | ||||
QFVE-3 | 0.1 | 3.2437E-02 | 6.5414E-03 | 1.6402E-03 | 4.1559E-04 | 1.0435E-04 | |
2.31000 | 1.99569 | 1.98067 | 1.99368 | ||||
0.2 | 3.2435E-02 | 6.5403E-03 | 1.6399E-03 | 4.1549E-04 | 1.0433E-04 | ||
2.31013 | 1.99576 | 1.98070 | 1.99372 | ||||
0.5 | 3.2428E-02 | 6.5373E-03 | 1.6389E-03 | 4.1523E-04 | 1.0425E-04 | ||
2.31047 | 1.99596 | 1.98076 | 1.99380 | ||||
0.9 | 3.2424E-02 | 6.5353E-03 | 1.6382E-03 | 4.1503E-04 | 1.0420E-04 | ||
2.31071 | 1.99617 | 1.98083 | 1.99389 | ||||
QFVE-4 | 0.1 | 4.3651E-02 | 1.2090E-02 | 3.1313E-03 | 7.9062E-04 | 1.9815E-04 | |
1.85224 | 1.94895 | 1.98570 | 1.99636 | ||||
0.2 | 4.3644E-02 | 1.2087E-02 | 3.1306E-03 | 7.9044E-04 | 1.9811E-04 | ||
1.85229 | 1.94896 | 1.98571 | 1.99637 | ||||
0.5 | 4.3624E-02 | 1.2080E-02 | 3.1288E-03 | 7.8997E-04 | 1.9798E-04 | ||
1.85246 | 1.94901 | 1.98573 | 1.99642 | ||||
0.9 | 4.3611E-02 | 1.2075E-02 | 3.1273E-03 | 7.8958E-04 | 1.9788E-04 | ||
1.85266 | 1.94907 | 1.98575 | 1.99646 |
Scheme | |||||||
---|---|---|---|---|---|---|---|
QFE | 0.1 | 3.8651E-02 | 4.9359E-03 | 5.9810E-04 | 7.3868E-05 | 9.2054E-06 | |
2.96912 | 3.04487 | 3.01736 | 3.00441 | ||||
0.2 | 3.8649E-02 | 4.9358E-03 | 5.9810E-04 | 7.3869E-05 | 9.2056E-06 | ||
2.96908 | 3.04485 | 3.01735 | 3.00438 | ||||
0.5 | 3.8644E-02 | 4.9356E-03 | 5.9809E-04 | 7.3869E-05 | 9.2061E-06 | ||
2.96897 | 3.04479 | 3.01731 | 3.00431 | ||||
0.9 | 3.8639E-02 | 4.9353E-03 | 5.9808E-04 | 7.3870E-05 | 9.2066E-06 | ||
2.96886 | 3.04472 | 3.01728 | 3.00424 | ||||
QFVE-1 | 0.1 | 3.5708E-02 | 4.7460E-03 | 5.9083E-04 | 7.3656E-05 | 9.2005E-06 | |
2.91146 | 3.00588 | 3.00388 | 3.00102 | ||||
0.2 | 3.5707E-02 | 4.7459E-03 | 5.9083E-04 | 7.3656E-05 | 9.2007E-06 | ||
2.91143 | 3.00586 | 3.00387 | 3.00100 | ||||
0.5 | 3.5703E-02 | 4.7457E-03 | 5.9083E-04 | 7.3657E-05 | 9.2011E-06 | ||
2.91135 | 3.00582 | 3.00385 | 3.00095 | ||||
0.9 | 3.5700E-02 | 4.7456E-03 | 5.9082E-04 | 7.3657E-05 | 9.2015E-06 | ||
2.91126 | 3.00578 | 3.00382 | 3.00089 | ||||
QFVE-2 | 0.1 | 3.6034E-02 | 4.9651E-03 | 6.6231E-04 | 9.9276E-05 | 1.8465E-05 | |
2.85945 | 2.90625 | 2.73799 | 2.42662 | ||||
0.2 | 3.6032E-02 | 4.9650E-03 | 6.6228E-04 | 9.9267E-05 | 1.8464E-05 | ||
2.85943 | 2.90628 | 2.73805 | 2.42657 | ||||
0.5 | 3.6028E-02 | 4.9646E-03 | 6.6219E-04 | 9.9241E-05 | 1.8460E-05 | ||
2.85940 | 2.90635 | 2.73824 | 2.42649 | ||||
0.9 | 3.6024E-02 | 4.9642E-03 | 6.6212E-04 | 9.9224E-05 | 1.8459E-05 | ||
2.85935 | 2.90638 | 2.73834 | 2.42633 | ||||
QFVE-3 | 0.1 | 3.5687E-02 | 6.8456E-03 | 1.6915E-03 | 4.2850E-04 | 1.0767E-04 | |
2.38217 | 2.01689 | 1.98093 | 1.99262 | ||||
0.2 | 3.5685E-02 | 6.8446E-03 | 1.6911E-03 | 4.2840E-04 | 1.0765E-04 | ||
2.38230 | 2.01697 | 1.98096 | 1.99265 | ||||
0.5 | 3.5679E-02 | 6.8417E-03 | 1.6901E-03 | 4.2813E-04 | 1.0757E-04 | ||
2.38265 | 2.01722 | 1.98103 | 1.99274 | ||||
0.9 | 3.5676E-02 | 6.8399E-03 | 1.6894E-03 | 4.2792E-04 | 1.0751E-04 | ||
2.38290 | 2.01746 | 1.98111 | 1.99282 | ||||
QFVE-4 | 0.1 | 4.4968E-02 | 1.2391E-02 | 3.2256E-03 | 8.1655E-04 | 2.0488E-04 | |
1.85956 | 1.94168 | 1.98197 | 1.99474 | ||||
0.2 | 4.4962E-02 | 1.2389E-02 | 3.2249E-03 | 8.1637E-04 | 2.0483E-04 | ||
1.85965 | 1.94171 | 1.98197 | 1.99476 | ||||
0.5 | 4.4946E-02 | 1.2382E-02 | 3.2231E-03 | 8.1588E-04 | 2.0471E-04 | ||
1.85991 | 1.94177 | 1.98200 | 1.99480 | ||||
0.9 | 4.4936E-02 | 1.2377E-02 | 3.2215E-03 | 8.1548E-04 | 2.0460E-04 | ||
1.86020 | 1.94185 | 1.98202 | 1.99484 |
Scheme | |||||||
---|---|---|---|---|---|---|---|
QFE | 0.1 | 4.1481E-02 | 5.2684E-03 | 6.6217E-04 | 8.2923E-05 | 1.0374E-05 | |
2.97701 | 2.99209 | 2.99736 | 2.99886 | ||||
0.2 | 4.1468E-02 | 5.2674E-03 | 6.6213E-04 | 8.2921E-05 | 1.0373E-05 | ||
2.97684 | 2.99190 | 2.99730 | 2.99888 | ||||
0.5 | 4.1435E-02 | 5.2646E-03 | 6.6202E-04 | 8.2916E-05 | 1.0372E-05 | ||
2.97645 | 2.99139 | 2.99715 | 2.99892 | ||||
0.9 | 4.1399E-02 | 5.2615E-03 | 6.6189E-04 | 8.2911E-05 | 1.0371E-05 | ||
2.97605 | 2.99081 | 2.99697 | 2.99894 | ||||
QFVE-1 | 0.1 | 4.0345E-02 | 5.2291E-03 | 6.6100E-04 | 8.2887E-05 | 1.0372E-05 | |
2.94777 | 2.98383 | 2.99544 | 2.99851 | ||||
0.2 | 4.0330E-02 | 5.2283E-03 | 6.6097E-04 | 8.2886E-05 | 1.0371E-05 | ||
2.94743 | 2.98367 | 2.99540 | 2.99852 | ||||
0.5 | 4.0287E-02 | 5.2261E-03 | 6.6089E-04 | 8.2882E-05 | 1.0371E-05 | ||
2.94649 | 2.98325 | 2.99528 | 2.99856 | ||||
0.9 | 4.0237E-02 | 5.2236E-03 | 6.6080E-04 | 8.2878E-05 | 1.0370E-05 | ||
2.94542 | 2.98276 | 2.99515 | 2.99858 | ||||
QFVE-2 | 0.1 | 4.3777E-02 | 5.9077E-03 | 8.3507E-04 | 1.4007E-04 | 2.9171E-05 | |
2.88949 | 2.82263 | 2.57578 | 2.26351 | ||||
0.2 | 4.3772E-02 | 5.9086E-03 | 8.3549E-04 | 1.4021E-04 | 2.9208E-05 | ||
2.88911 | 2.82213 | 2.57507 | 2.26313 | ||||
0.5 | 4.3761E-02 | 5.9117E-03 | 8.3684E-04 | 1.4065E-04 | 2.9327E-05 | ||
2.88800 | 2.82056 | 2.57281 | 2.26184 | ||||
0.9 | 4.3748E-02 | 5.9151E-03 | 8.3830E-04 | 1.4113E-04 | 2.9457E-05 | ||
2.88675 | 2.81887 | 2.57042 | 2.26038 | ||||
QFVE-3 | 0.1 | 3.8981E-02 | 1.1306E-02 | 3.0314E-03 | 7.7277E-04 | 1.9412E-04 | |
1.78570 | 1.89902 | 1.97187 | 1.99310 | ||||
0.2 | 3.8977E-02 | 1.1322E-02 | 3.0363E-03 | 7.7408E-04 | 1.9445E-04 | ||
1.78356 | 1.89867 | 1.97178 | 1.99306 | ||||
0.5 | 3.8977E-02 | 1.1372E-02 | 3.0521E-03 | 7.7824E-04 | 1.9552E-04 | ||
1.77714 | 1.89759 | 1.97153 | 1.99292 | ||||
0.9 | 3.8967E-02 | 1.1425E-02 | 3.0690E-03 | 7.8269E-04 | 1.9665E-04 | ||
1.77006 | 1.89635 | 1.97125 | 1.99279 | ||||
QFVE-4 | 0.1 | 8.0001E-02 | 2.2742E-02 | 5.8901E-03 | 1.4858E-03 | 3.7223E-04 | |
1.81469 | 1.94898 | 1.98708 | 1.99694 | ||||
0.2 | 8.0055E-02 | 2.2776E-02 | 5.9000E-03 | 1.4883E-03 | 3.7288E-04 | ||
1.81345 | 1.94875 | 1.98702 | 1.99691 | ||||
0.5 | 8.0248E-02 | 2.2887E-02 | 5.9314E-03 | 1.4964E-03 | 3.7492E-04 | ||
1.80993 | 1.94811 | 1.98686 | 1.99683 | ||||
0.9 | 8.0435E-02 | 2.3005E-02 | 5.9650E-03 | 1.5051E-03 | 3.7712E-04 | ||
1.80590 | 1.94733 | 1.98667 | 1.99675 |
Scheme | |||||||
---|---|---|---|---|---|---|---|
QFE | 0.1 | 4.5775E-02 | 5.8108E-03 | 7.2556E-04 | 9.0654E-05 | 1.1335E-05 | |
2.97775 | 3.00156 | 3.00067 | 2.99964 | ||||
0.2 | 4.5752E-02 | 5.8105E-03 | 7.2556E-04 | 9.0653E-05 | 1.1334E-05 | ||
2.97711 | 3.00149 | 3.00066 | 2.99967 | ||||
0.5 | 4.5696E-02 | 5.8099E-03 | 7.2556E-04 | 9.0652E-05 | 1.1333E-05 | ||
2.97549 | 3.00134 | 3.00068 | 2.99976 | ||||
0.9 | 4.5639E-02 | 5.8094E-03 | 7.2557E-04 | 9.0652E-05 | 1.1333E-05 | ||
2.97381 | 3.00121 | 3.00070 | 2.99984 | ||||
QFVE-1 | 0.1 | 4.4883E-02 | 5.7726E-03 | 7.2486E-04 | 9.0665E-05 | 1.1336E-05 | |
2.95887 | 2.99345 | 2.99908 | 2.99961 | ||||
0.2 | 4.4848E-02 | 5.7724E-03 | 7.2486E-04 | 9.0665E-05 | 1.1336E-05 | ||
2.95779 | 2.99339 | 2.99908 | 2.99964 | ||||
0.5 | 4.4755E-02 | 5.7720E-03 | 7.2488E-04 | 9.0666E-05 | 1.1335E-05 | ||
2.95492 | 2.99325 | 2.99911 | 2.99972 | ||||
0.9 | 4.4658E-02 | 5.7717E-03 | 7.2491E-04 | 9.0667E-05 | 1.1335E-05 | ||
2.95186 | 2.99312 | 2.99915 | 2.99979 | ||||
QFVE-2 | 0.1 | 4.8680E-02 | 6.5149E-03 | 9.1633E-04 | 1.5331E-04 | 3.1883E-05 | |
2.90151 | 2.82980 | 2.57944 | 2.26557 | ||||
0.2 | 4.8643E-02 | 6.5161E-03 | 9.1675E-04 | 1.5344E-04 | 3.1917E-05 | ||
2.90016 | 2.82941 | 2.57887 | 2.26527 | ||||
0.5 | 4.8549E-02 | 6.5202E-03 | 9.1810E-04 | 1.5386E-04 | 3.2029E-05 | ||
2.89644 | 2.82819 | 2.57703 | 2.26419 | ||||
0.9 | 4.8447E-02 | 6.5248E-03 | 9.1956E-04 | 1.5431E-04 | 3.2148E-05 | ||
2.89240 | 2.82692 | 2.57511 | 2.26301 | ||||
QFVE-3 | 0.1 | 4.2879E-02 | 1.2323E-02 | 3.3070E-03 | 8.4521E-04 | 2.1258E-04 | |
1.79895 | 1.89770 | 1.96816 | 1.99132 | ||||
0.2 | 4.2956E-02 | 1.2337E-02 | 3.3117E-03 | 8.4645E-04 | 2.1290E-04 | ||
1.79993 | 1.89732 | 1.96805 | 1.99127 | ||||
0.5 | 4.3189E-02 | 1.2382E-02 | 3.3265E-03 | 8.5044E-04 | 2.1392E-04 | ||
1.80242 | 1.89616 | 1.96773 | 1.99112 | ||||
0.9 | 4.3451E-02 | 1.2429E-02 | 3.3422E-03 | 8.5467E-04 | 2.1501E-04 | ||
1.80572 | 1.89482 | 1.96735 | 1.99097 | ||||
QFVE-4 | 0.1 | 8.5404E-02 | 2.4749E-02 | 6.4398E-03 | 1.6279E-03 | 4.0823E-04 | |
1.78695 | 1.94227 | 1.98398 | 1.99559 | ||||
0.2 | 8.5576E-02 | 2.4780E-02 | 6.4492E-03 | 1.6304E-03 | 4.0885E-04 | ||
1.78801 | 1.94201 | 1.98390 | 1.99556 | ||||
0.5 | 8.6107E-02 | 2.4883E-02 | 6.4795E-03 | 1.6383E-03 | 4.1086E-04 | ||
1.79094 | 1.94123 | 1.98370 | 1.99548 | ||||
0.9 | 8.6678E-02 | 2.4991E-02 | 6.5118E-03 | 1.6467E-03 | 4.1300E-04 | ||
1.79424 | 1.94030 | 1.98345 | 1.99539 |
Scheme | |||||||
---|---|---|---|---|---|---|---|
QFE | 0.1 | 4.9259E −04 | 6.2585E −05 | 7.8691E −06 | 9.8500E −07 | 1.2318E −07 | |
2.97649 | 2.99155 | 2.99801 | 2.99930 | ||||
0.2 | 5.2389E −04 | 6.6536E −05 | 8.3568E −06 | 1.0456E −06 | 1.3085E −07 | ||
2.97704 | 2.99312 | 2.99864 | 2.99831 | ||||
0.5 | 6.1113E −04 | 7.7927E −05 | 9.7900E −06 | 1.2247E −06 | 1.5342E −07 | ||
2.97128 | 2.99275 | 2.99892 | 2.99680 | ||||
0.9 | 6.5735E −04 | 8.3896E −05 | 1.0542E −05 | 1.3187E −06 | 1.6523E −07 | ||
2.96999 | 2.99251 | 2.99892 | 2.99658 | ||||
QFVE −1 | 0.1 | 4.9297E −04 | 6.2703E −05 | 7.8750E −06 | 9.8522E −07 | 1.2319E −07 | |
2.97488 | 2.99319 | 2.99876 | 2.99953 | ||||
0.2 | 5.2388E −04 | 6.6592E −05 | 8.3597E −06 | 1.0457E −06 | 1.3086E −07 | ||
2.97582 | 2.99383 | 2.99899 | 2.99842 | ||||
0.5 | 6.1107E −04 | 7.7931E −05 | 9.7902E −06 | 1.2247E −06 | 1.5342E −07 | ||
2.97108 | 2.99278 | 2.99895 | 2.99680 | ||||
0.9 | 6.5735E −04 | 8.3896E −05 | 1.0542E −05 | 1.3187E −06 | 1.6523E −07 | ||
2.96999 | 2.99251 | 2.99892 | 2.99658 | ||||
QFVE −2 | 0.1 | 4.9668E −04 | 6.3296E −05 | 8.0014E −06 | 1.0243E −06 | 1.3406E −07 | |
2.97211 | 2.98380 | 2.96568 | 2.93362 | ||||
0.2 | 5.2619E −04 | 6.6948E −05 | 8.4312E −06 | 1.0668E −06 | 1.3666E −07 | ||
2.97446 | 2.98923 | 2.98250 | 2.96460 | ||||
0.5 | 6.1138E −04 | 7.7973E −05 | 9.7965E −06 | 1.2260E −06 | 1.5373E −07 | ||
2.97104 | 2.99262 | 2.99832 | 2.99551 | ||||
0.9 | 6.5736E −04 | 8.3897E −05 | 1.0542E −05 | 1.3187E −06 | 1.6523E −07 | ||
2.96999 | 2.99251 | 2.99892 | 2.99658 | ||||
QFVE −3 | 0.1 | 4.8358E −04 | 6.6170E −05 | 1.0897E −05 | 2.2481E −06 | 4.3215E −07 | |
2.86950 | 2.60230 | 2.27712 | 2.37910 | ||||
0.2 | 5.1682E −04 | 6.8178E −05 | 9.7474E −06 | 1.7606E −06 | 3.2209E −07 | ||
2.92227 | 2.80622 | 2.46895 | 2.45052 | ||||
0.5 | 6.0951E −04 | 7.7849E −05 | 9.8406E −06 | 1.2575E −06 | 1.6444E −07 | ||
2.96890 | 2.98385 | 2.96824 | 2.93488 | ||||
0.9 | 6.5732E −04 | 8.3893E −05 | 1.0541E −05 | 1.3187E −06 | 1.6523E −07 | ||
2.96999 | 2.99250 | 2.99890 | 2.99654 | ||||
QFVE −4 | 0.1 | 5.0009E −04 | 8.2417E −05 | 1.7586E −05 | 4.0042E −06 | 7.9228E −07 | |
2.60119 | 2.22855 | 2.13480 | 2.33744 | ||||
0.2 | 5.2385E −04 | 7.5028E −05 | 1.3824E −05 | 2.9577E −06 | 5.7507E −07 | ||
2.80366 | 2.44027 | 2.22464 | 2.36264 | ||||
0.5 | 6.0883E −04 | 7.8074E −05 | 1.0018E −05 | 1.3420E −06 | 1.9249E −07 | ||
2.96311 | 2.96223 | 2.90012 | 2.80159 | ||||
0.9 | 6.5730E −04 | 8.3890E −05 | 1.0541E −05 | 1.3187E −06 | 1.6524E-07 | ||
2.96998 | 2.99249 | 2.99887 | 2.99647 |
The exact solution to this example is:
Analogously, we calculate the
We take the space-time domain
In numerical calculation of this example, we use
In the last example, we choose the nonlinear term
In this article, we study a nonlinear time-fractional Rayleigh-Stokes problem by using the quadratic finite volume element method combined with a specific time discretization. In temporal direction, we use a two step scheme to approximate the equation at time
The authors would like to thank the editor and the anonymous reviewers for their valuable suggestions.