1 Introduction
2 Equations and Schemes
2.1 Time Stepping
2.1.1 Backward Euler
2.1.2 Eyre’s Method
2.1.3 Second Order Schemes
2.2 Spatial Discretization and Solution Procedure
2.3 Error Estimation and Adaptive Time Stepping
3 Local Truncation Errors in Metastable Dynamics
3.1 Metastable Dynamics
3.1.1 Predicted Time Step Sizes for AC
Method (AC) |
L
|
k
|
\(M = O(1/k)\)
|
---|---|---|---|
BE |
\(k^2/\epsilon ^2\)
|
\(\sqrt{\sigma }\epsilon \)
|
\(1/(\sqrt{\sigma }\epsilon )\)
|
Eyre, IMEX1, SAV1 |
\(k^2/\epsilon ^3\)
|
\(\sqrt{\sigma }\epsilon ^{3/2}\)
|
\(1/(\sqrt{\sigma }\epsilon ^{3/2})\)
|
TR, BDF2 |
\(k^3/\epsilon ^3\)
|
\(\root 3 \of {\sigma }\epsilon \)
|
\(1/(\root 3 \of {\sigma }\epsilon )\)
|
S, DIRK2, SBDF2, SAV2-A |
\(k^3/\epsilon ^4\)
|
\(\root 3 \of {\sigma }\epsilon ^{4/3}\)
|
\(1/(\root 3 \of {\sigma }\epsilon ^{4/3})\)
|
SAV2-B |
\(k^3/\epsilon ^5\)
|
\(\root 3 \of {\sigma }\epsilon ^{5/3}\)
|
\(1/(\root 3 \of {\sigma }\epsilon ^{5/3})\)
|
3.1.2 Predicted Time Step Sizes for CH
Method (CH) | L | k | \(M = O(1/k)\) |
---|---|---|---|
BE | \(k^2/\epsilon ^2\) | \(\sqrt{\sigma }\epsilon \) | \(1/(\sqrt{\sigma }\epsilon )\) |
Eyre, IMEX1, SAV1 | \(k^2/\epsilon ^4\) | \(\sqrt{\sigma }\epsilon ^2\) | \(1/(\sqrt{\sigma }\epsilon ^2)\) |
TR, BDF2 | \(k^3/\epsilon ^3\) | \(\root 3 \of {\sigma }\epsilon \) | \(1/(\root 3 \of {\sigma }\epsilon )\) |
S, DIRK2, SBDF2, SAV2-A | \(k^3/\epsilon ^5\) | \(\root 3 \of {\sigma }\epsilon ^{5/3}\) | \(1/(\root 3 \of {\sigma }\epsilon ^{5/3})\) |
3.1.3 Discussion: The Source of Increased Local Error
4 Computational Results
4.1 Allen–Cahn
4.1.1 First Order Methods
\(\sigma \) | BE | Eyre | ||||
---|---|---|---|---|---|---|
M | CG | E | M | CG | E | |
1e−4 | 717 | 5348 [7.46] | 0.003 | 2,350 | 14,856 [6.32] | 0.047 |
1e−5 | 2225 (3.10) | 9448 [4.24] | 0.001 | 7351 (3.12) | 28,263 [3.85] | 0.014 |
1e−6 | 7010 (3.15) | 23,017 [3.28] | 0.001 | 23,172 (3.15) | 68,148 [2.94] | 0.004 |
\(\epsilon \) | BE | Eyre | ||||
---|---|---|---|---|---|---|
M | CG | E | M | CG | E | |
0.2 | 717 | 5,348 [7.46] | 0.003 | 2350 | 14,856 [6.32] | 0.047 |
0.1 | 1291 (1.80) | 12,354 [9.57] | 0.001 | 6463 (2.75) | 44,717 [6.92] | 0.069 |
0.05 | 2412 (1.87) | 27,782 [11.52] | 0.001 | 18,218 (2.83) | 143,416 [7.87] | 0.099 |
0.025 | 4630 (1.92) | 64,884 [14.01] | \(*\) | 52,595 (2.89) | 497,846 [9.47] | 0.141 |
\(\epsilon \) | IMEX1 | SAV1 | ||
---|---|---|---|---|
M | E | M | E | |
0.2 | 3932 | 0.067 | 3936 | 0.067 |
0.1 | 11,110 (2.83) | 0.096 | 11,112 (2.82) | 0.096 |
0.05 | 31,676 (2.85) | 0.138 | 31,682 (2.85) | 0.138 |
0.025 | 90,748 (2.86) | 0.198 | 90,760 (2.86) | 0.198 |
4.1.2 Second Order Methods
\(\epsilon \) | TR | S | BDF2 | DIRK2 | SBDF2 | SAV2-A | SAV2-B |
---|---|---|---|---|---|---|---|
0.2 | 170 | 236 | 280 | 180 | 588 | 768 | 1,572 |
0.1 | 278 (1.64) | 512 (2.16) | 472 (1.69) | 364 (2.02) | 1384 (2.35) | 1572 (2.04) | 5436 (3.46) |
0.05 | 492 (1.77) | 1208 (2.36) | 860 (1.82) | 814 (2.24) | 3260 (2.35) | 3392 (2.16) | 15,088 (2.78) |
0.025 | 916 (1.86) | 2,960 (2.45) | 1632 (1.90) | 1894 (2.33) | 7600 (2.33) | 7980 (2.35) | 48,048 (3.18) |
4.2 Cahn–Hilliard
4.2.1 First Order Methods
\(\epsilon \) | BE | Eyre | IMEX1 | |||||
---|---|---|---|---|---|---|---|---|
M | CG | E | M | CG | E | M | E | |
0.2 | 730 | 5348 [7.33] | \(*\) | 3055 | 36,684 [12.0] | 0.019 | 9982 | 0.014 |
0.1 | 1184 (1.62) | 24,778 [20.9] | 0.001 | 12,751 (4.17) | 190,204 [14.0] | 0.021 | 43,332 (0.015) | 0.015 |
0.05 | 2068 (1.75) | 66,307 [32.1] | \(*\) | 52,753 (4.13) | 937,774 [17.8] | 0.022 | 181,234 (4.18) | 0.015 |
0.025 | 3768 (1.82) | 198,771 [52.8] | \(*\) | 215,443 (4.08) | 4,504,278 [20.9] | 0.022 | 740,366 (4.09) | 0.015 |
4.2.2 Second Order Methods
\(\epsilon \) | TR | S | BDF2 | DIRK2 | SBDF2 |
---|---|---|---|---|---|
0.2 | 230 | 534 | 320 | 378 | 1388 |
0.1 | 314 (1.36) | 1530 (2.87) | 468 (1.46) | 788 (2.08) | 4108 (2.96) |
0.05 | 474 (1.51) | 4722 (3.08) | 748 (1.60) | 1906 (2.42) | 12,352 (3.01) |
0.025 | 792 (1.67) | 14,924 (3.16) | 1312 (1.75) | 6048 (3.17) | 44,060 (3.57) |
5 Asymptotic Analysis of Properties of BE AC Solutions
6 Rigorous Radial Analysis of AC With BE and Eyre Time Stepping
6.1 Backward Euler Estimates
6.2 Eyre-Type Iterations
6.2.1 Computational Validation of Remark 7
\(\epsilon \) | Eyre with \(f(u) = u^5-u^3\) | |
---|---|---|
M | E | |
0.2 | 5726 | 0.001 |
0.1 | 21,947 (3.83) | 0.005 |
0.05 | 86,499 (3.94) | 0.007 |
0.025 | 343,525(3.97) | 0.007 |