One-Dimensional Shallow Water Equations Ill-Posedness
Abstract
1. Introduction
2. Mathematical Background
2.1. Governing Equations
2.2. Evolution of Internal Perturbations
2.3. Principle of Frozen Coefficients
2.4. Well-Posed Problems
- Problem (20) is considered well-posed if it possesses a unique smooth solution and exhibits stability.
- Problem (20) is stable if there are constants , and , such that:
2.5. Fourier Transform Problem
3. Methodology
4. Results
4.1. Fourier Transform Solution
4.2. Ill-Posedness Condition
5. Discussion
5.1. Friction Slope
5.2. Physical Interpretation
5.3. Numerical Methods
5.4. Roll Waves
6. Conclusions
- The 1D SWE are ill-posed if the following condition is satisfied:Thus, , the essential criterion for the emergence of roll waves in uniform flow, is also applicable for unsteady flow when the 1D SWE become ill-posed.
- 2.
- Condition (63) indicates that the 1D SWE become ill-posed, meaning they cannot yield any solution, regardless of the numerical scheme or method employed.
- 3.
- The challenges in simulating transcritical flow stem from the ill-posedness of the 1D SWE, rather than the Preissmann numerical scheme, as previously suggested by various researchers over the years. If the 1D SWE are well-posed (i.e., condition (63) is not satisfied), any stable numerical scheme or method can effectively simulate transcritical flows.
- 4.
- For a given prismatic channel, the maximum Froude number at which the 1D SWE remains solvable is determined by regardless of the numerical method used. Specifically, for very wide channels, the 1D SWE remain valid for , under the Manning equation for the friction slope, while the Chézy equation allows validity for .
- 5.
- Roll waves formation is revisited in this paper, and a complete procedure to solve the 1D SWE with roll waves formation is provided.
- 6.
- The uniform flow is not possible when the ill-posedness condition is not met. This is an important result, as the design of channels with high slopes is based on uniform flow concept.
- 7.
- Finally, the findings regarding the ill-posedness of the 1D SWE discussed in this paper will enhance the quality of open-channel flow education at both undergraduate and graduate levels, ultimately contributing to the development of well-trained professionals in the field. The Vedernikov number should be integrated into open-channel flow curricula alongside the Froude number.
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
1D SWE | One Dimensional Shallow Water Equations |
A | wetted area (m2) |
B | bottom width of a trapezoidal channel (m) |
Chézy coefficient (m1/2/s) | |
D = A/T | hydraulic depth (m) |
Froude number (-) | |
constants in stability definition | |
LPI | Local Partial Inertia |
M | (-) |
matrix of the perturbed 1D SWE | |
PDE | Partial Differential Equation |
R = A/P | hydraulic radius (m) |
S0 | bed slope (-) |
Sf | friction slope (-) |
perturbed friction slope (-) | |
source term vector of the 1D SWE | |
source term vector of the perturbation 1D SWE | |
source term vector of the perturbed 1D SWE | |
flow field of the perturbed 1D SWE | |
initial flow field of the 1D SWE | |
initial perturbation field of the perturbed 1D SWE | |
T | top water surface width (m) |
Vedernikov number (-) | |
in stability definition | |
function in stability definition | |
g | gravity acceleration (m/s2) |
h | water depth (m) |
depth’s perturbation (m) | |
, | |
Manning coefficient (s/m1/3) | |
t | time independent variable (s) |
u | flow velocity (m/s) |
velocity’s perturbation (m/s) | |
eigenvectors | |
x | spatial independent variable (m) |
z | slope side, V:H = 1/z |
dimensionless depth (-) | |
Elements of the Jacobian matrix of Sf | |
constants in stability definition | |
weighting coefficient of the Preissmann’s scheme (-) | |
parameter of the LPI technique (-) | |
wave number (m−1), in the definition of Fourier transform |
Appendix A. Evolution of Internal Perturbation
Appendix B. Fourier Transform Pairs
Appendix C. Matrix Exponential
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Mahdi, T.-F. One-Dimensional Shallow Water Equations Ill-Posedness. Mathematics 2025, 13, 2476. https://doi.org/10.3390/math13152476
Mahdi T-F. One-Dimensional Shallow Water Equations Ill-Posedness. Mathematics. 2025; 13(15):2476. https://doi.org/10.3390/math13152476
Chicago/Turabian StyleMahdi, Tew-Fik. 2025. "One-Dimensional Shallow Water Equations Ill-Posedness" Mathematics 13, no. 15: 2476. https://doi.org/10.3390/math13152476
APA StyleMahdi, T.-F. (2025). One-Dimensional Shallow Water Equations Ill-Posedness. Mathematics, 13(15), 2476. https://doi.org/10.3390/math13152476