A Five-Step Block Method Coupled with Symmetric Compact Finite Difference Scheme for Solving Time-Dependent Partial Differential Equations
Abstract
:1. Introduction
2. Development of a Five-Step Block Method
3. Basic Characteristics of the Method and Stability Analysis
3.1. Order of Accuracy and Consistency
3.2. Zero Stability
3.3. Linear Stability Analysis
4. Symmetric Compact Finite Difference Scheme
5. Test Problems
5.1. Burgers’ Equation
5.2. FitzHugh–Nagumo Equation
5.3. Stability of Differential System
6. Numerical Experiments
6.1. Nonlinear Burgers’ Equation
6.1.1. Example 1
6.1.2. Example 2
6.2. Non-Linear FitzHugh–Nagumo Equation
6.2.1. Example 1
6.2.2. Example 2
6.3. PDE with Manufactured Solution
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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x | Absolute Error |
---|---|
0.1 | 4.6342 × |
0.2 | 1.4033 × |
0.3 | 1.9936 × |
0.4 | 1.2330 × |
0.5 | 4.9473 × |
0.6 | 2.3491 × |
0.7 | 3.8157 × |
0.8 | 6.0069 × |
0.9 | 1.5378 × |
x | Absolute Error (Asai [31]) | Absolute Error (Mittal [20]) | Absolute Error (The Proposed Scheme) |
---|---|---|---|
0.1 | 4.50 × | 7.40 × | 3.84 × |
0.2 | 7.70 × | 6.00 × | 5.45 × |
0.3 | 1.21 × | 1.20 × | 6.79 × |
0.4 | 2.40 × | 1.78 × | 1.23 × |
0.5 | 2.53 × | 3.90 × | 4.41 × |
0.6 | 3.56 × | 4.40 × | 1.63 × |
0.7 | 4.84 × | 1.00 × | 2.53 × |
0.8 | 3.32 × | 7.40 × | 3.48 × |
0.9 | 4.17 × | 2.81 × | 7.86 × |
x | Absolute Error (Asai [31]) | Absolute Error (Mittal [20]) | Absolute Error (The Proposed Scheme) |
---|---|---|---|
0.1 | 4.00 × | 0.000000 | 5.76 × |
0.2 | 9.00 × | 2.00 × | 1.47 × |
0.3 | 1.40 × | 3.00 × | 1.96 × |
0.4 | 2.20 × | 6.00 × | 2.82 × |
0.5 | 3.20 × | 1.00 × | 9.95 × |
0.6 | 4.90 × | 1.20 × | 4.02 × |
0.7 | 7.50 × | 7.50 × | 6.72 × |
0.8 | 4.50 × | 1.00 × | 1.58 × |
0.9 | 8.10 × | 7.40 × | 3.35 × |
N | -Error | ROC |
---|---|---|
40 | 8.08777 × | |
80 | 9.97993 × | 3.0186 |
160 | 3.83759 × | 4.7008 |
320 | 1.69230 × | 4.5031 |
N | -Error with CPU (sec.) (The Proposed Scheme) | (Method in Akkoyunlu [9]) |
---|---|---|
12 | 6.3673 × | 3.9857 × |
(0.65) | ||
24 | 4.7712 × | 2.3475 × |
(2.34) | ||
48 | 8.0504 × | 8.3749 × |
(9.60) | ||
64 | 4.7818 × | 5.9363 × |
(17.75) | ||
No. of iterations | 4 | 20 |
Ahmad [32] | Jiwari [11] | The Proposed Scheme | |
---|---|---|---|
-Errorh | -Error | -Error with CPU (sec.) | |
0.2 | 2.1960 × | 1.5880 × | 4.0099 × |
(47.89) | |||
0.5 | 1.5696 × | 3.8433 × | 2.8629 × |
(125.23) | |||
1.0 | 7.1449 × | 8.1870 × | 1.3175 × |
(268.56) | |||
1.5 | 1.7262 × | 1.3387 × | 3.2282 × |
(395.18) | |||
2 | 3.1857 × | 1.9433 × | 6.1114 × |
(527.95) | |||
time step-size(k) | 0.0001 | 0.001 | 0.01 |
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Kaur, K.; Singh, G.; Ritelli, D. A Five-Step Block Method Coupled with Symmetric Compact Finite Difference Scheme for Solving Time-Dependent Partial Differential Equations. Symmetry 2024, 16, 307. https://doi.org/10.3390/sym16030307
Kaur K, Singh G, Ritelli D. A Five-Step Block Method Coupled with Symmetric Compact Finite Difference Scheme for Solving Time-Dependent Partial Differential Equations. Symmetry. 2024; 16(3):307. https://doi.org/10.3390/sym16030307
Chicago/Turabian StyleKaur, Komalpreet, Gurjinder Singh, and Daniele Ritelli. 2024. "A Five-Step Block Method Coupled with Symmetric Compact Finite Difference Scheme for Solving Time-Dependent Partial Differential Equations" Symmetry 16, no. 3: 307. https://doi.org/10.3390/sym16030307
APA StyleKaur, K., Singh, G., & Ritelli, D. (2024). A Five-Step Block Method Coupled with Symmetric Compact Finite Difference Scheme for Solving Time-Dependent Partial Differential Equations. Symmetry, 16(3), 307. https://doi.org/10.3390/sym16030307