On the approximation of dam-break problems using a fuzzified HR-TVD scheme

Graphical abstract

Abstract

The construction of the proposed second-order accurate scheme is based on the Monotonic Upstream Schemes for Conservation Laws (MUSCL) strategy. Using some strong notions from fuzzy logic, like fuzzy modifiers and fuzzy inference systems, the work offers a new avenue for optimizing the execution of classical flux-limiting algorithms. A numerical study has also been included to compare the new limiting scheme and the classical Monotonized Central (MC) scheme. The key themes of the research work are:

• •Development of the background theory and augmented Riemann finite volume (FV) framework required to construct the new fuzzy logic-based limiter for the Shallow Water Equations (SWEs) over flat bottom topography.
• •The proposed fuzzy flux limiting technique has been employed in a structured one-dimensional FV model to compute the SWEs.
• •The idealized dam-break flows verify the new fuzzified High-Resolution (HR), Total Variation Diminishing (TVD) technique.

Specifications table

Subject area:	Mathematics and Statistics
More specific subject area:	Hyperbolic Conservation Laws
Name of your method:	Fuzzy flux limiting numerical methods
Name and reference of the original method:	N.A.
Resource availability:	N.A.

Introduction

The SWEs are the partial differential equations (PDEs) exhibiting a hyperbolic nature. Such equations are frequently used in various applications such as earthquakes, ocean currents, floods, reservoirs flows, and many more [1], [2], [3], [4], [5], [6], [7]. The one-dimensional SWEs represent shallow water dam-break flows, a non-linear coupled system of hyperbolic PDEs [8], [9], [10]. The shallow water models have been used repeatedly in plenty of fields, and some frequently occurring examples are irrigation flows, drainage systems, floods, and tidal flows [11], [12]. In recent years significant efforts have been made to model one-dimensional open channel shallow water dam-break flow problems [8], [11], [13]. Quite a range of computational methods have been proposed and successfully applied to solve the SWEs [14], [15], [16]. The main idea is based upon the concept of solving a series of Riemann problems for the fluid at each time step to perform cell averaging. But, this becomes a tedious process, and it becomes impossible for some non-linear cases [9], [10], [17]. While LeVeque has certainly done great things for the numerical solution of hyperbolic PDEs, the idea of using Riemann solvers dates back to Godunov [18]. The Riemann solvers for solving the shallow water equations are indeed quite robust and mature [19], [20].

Dam-break events are related to major ecological problems triggered by the rapid release of reservoir-stored water. The actual wave structure observed experimentally varies in large parts of the wave profile, such as the negative and positive fronts. Three specific problems occur in the numerical simulation of the shallow water models for various wave propagation phenomena over real domains [21]. The first issue is the approximation of the fronts or abrupt waves of fluid that could be interpreted as a numerically propagating discontinuity [19], [22], [23]. The second issue stems from sudden alterations in bathymetry (the measurement of the depth of water in water bodies). Most simulation methods offer a practical approach to the flow as long as the bottom surface remains relatively smooth [3]. In addition, the third issue occurs when these systems are implemented to examine the propagation of the wavefront over a dry bed. Some challenges are to be tackled during the simulations of such a work, involving the dam-break problems; hence, the shallow water dam-break models are worth a vital concern. In general, it is challenging to find solutions to these equations due to the hyperbolic behavior of the SWEs. These issues are robustly triggered by implementing some modern shock-capturing strategies to compute the shallow water dam-break flow systems.

The main interest in this work is to formulate the fuzzy logic-based numerical technique for the Shallow water system and its variants for flows over flat topography and cross-section using FV flux-limited methods [1], [24], [25]. The proposed fuzzy flux limiting technique has been employed in a structured one-dimensional FV model to compute the SWEs [24], [26], [27]. This work is leaned towards the computation of one dimensional SWEs using some robust concepts from the fuzzy logic theory. In the next sections, these objectives are systematically documented. The paper is now structured in the following manner.

• •The mathematics of the flow model including the governing equations, discretization techniques, and the numerical approach has been mentioned in the Section “Numerical model”.
• •In the Section “The new flux-limiter scheme”, the new FV based numerical technique with a brief discussion on fuzzy logic concepts has been discussed for a rectangular domain.
• •After that, a numerical assessment is shown in the Section “Numerical simulation and results”. For validation purpose, a comparison of the proposed limiter and the classic MC limiter has been presented here.
• •The information about the notations used in this work is given in the Section “Notations”.
• •Furthermore, in the Section “Discussion and remarks”, the work has been concluded with several observations and the scope of future studies.

Numerical model

The Navier Stokes equations are employed to study the behavior of fluids. The behavior of a fluid in shallow areas is defined by these equations. Despite the strong assumptions used during approximations, the outcomes are comparable to reality, even in cases where many of these assumptions aren’t really fulfilled. Among various solvable problems are sea currents, tidal flows, flow in channels, or propagation of shock waves. The one-dimensional version of these Navier-stokes equations leads to the SWEs under particular assumptions. But, despite considerable simplifications, the Navier Stokes equations have no analytical solution, even in one-dimensional SWEs. Thus, approximation methods are imposed to simulate such non-linear systems. The numerical model in this work is about the flow pattern formed due to the collapse of a dam, which can be expressed mathematically by the SWEs [13].

The governing equations

By assuming negligible vertical velocity, a constant horizontal velocity, and a constant density, the Navier-Stokes equations become the SWEs for an in-compressible fluid. The vertical velocity is thus ignored for the dam break situations. The horizontal velocity is considered constant across any vertical line between the base and the surface of the reservoir. (as shown in the Fig. 1) By performing vertical integration to the Navier Stokes equations, the resultant are the SWEs. The velocity and gradients in the horizontal direction parallel to the dam also are overlooked in the dam break problems. Thus, in one space dimension, the SWEs [17] in the conservative form for the rectangular domain of unit width, can be presented via the following equation:

$$q_t+f{\left(q\right)}_x=s,$$ (1)

where $q$, $f$, and $s$ vectors are as follows:

$$q=\left(\begin{array}{c}h\\ hu\end{array}\right)\text{,}f\left(q\right)=\left(\begin{array}{c}hu\\ h{u}^2+1/2g{h}^2\alpha \end{array}\right)\text{,}s\left(q\right)=\left(\begin{array}{c}0\\ -gh{b}_x\end{array}\right).$$

Here, $h(x,t)$ represents the height of the water, $u(x,t)$ is the fluid velocity, $g$ is the notation for gravitational constant and the function of bottom topography is represented by $b\left(x\right)$. As, this research is mainly concentrated towards hyperbolic conservation laws, hence the function $b\left(x\right)$ has been taken as zero. So, the Eq. (1) becomes:

$${\partial }_t\left(\begin{array}{c}h\\ hu\end{array}\right)+{\partial }_x\left(\begin{array}{c}hu\\ h{u}^2+\frac{1}{2}g{h}^2\end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right).$$ (2)

The FV paradigm has been chosen in this study because it avoids any global alterations in the conserved components, ensuring that the entire scheme is conservative. As shown in the subsequent sections, both spatial and time discretizations are handled in a higher-order accurate manner.

Fig. 1.

A graphical representation of shallow water model, here $\eta$ is the manning’s roughness coefficient [28].

Discretization technique

For the spatial discretization, the aforementioned three techniques, namely the FDMs, FEMs, and FVMs, could be used for the shallow water dam-break numerical simulations [29], [30]. The Finite-Difference Approach is the oldest, but, possibly because of its lack of versatility from the geometric point of view, its popularity has steadily declined. For this work, the FV formulation is employed since the development of numerical algorithms in the FV frameworks needs less computational work also these are suitable for complex geometries, both structured and unstructured [31], [32], [33].

Numerical approach based on the FV framework

A spatial reconstruction, called the MUSCL scheme, has been used for achieving high order accuracy in space. To formulate the FV framework, the key task is the discretization of the spatial domain into smaller nodes $[x_{i-1/2},x_{i+1/2}]$, having a uniform length $\Delta x$ in space component, hence $x_i=\Delta x(i+1/2)$. In same manner, the time interval is discretized into sub divisions $[t^n,t^{n+1}]$ with a same time step of length $\Delta t$. Thus, $t^n=\left(\Delta t\right)n$. $[x_{i-1/2},x_{i+1/2}]$ represents the $i$th computational volume, where $x_i=(i+1/2)\Delta x$ is the midpoint of this cell. The initial fluid interface pattern and rate of the fluid flow should be defined and followed along-with the domain of the channel for the approximation of unsteady flow systems before the beginning of the computational process.

An approximation of the non-linear hyperbolic conservation laws, presented in the Section “The governing equations” requires a FV Godunov method of upwind-type. The FV solutions of the Eq. (2) for capturing the shocks by implementing the Godunov’s upwind technique, and some robust flux limiting schemes, lead to accurate simulation results of the considered shallow water dam break systems [3], [18], [34]. The integral form of the Eq. (1) over a uniform mesh in the x-t plane is:

$${\int }_t^{t+\Delta t}{\int }_{x_{i-1/2}}^{x_{i+1/2}}\frac{\partial \mathbf{q}}{\partial t}\mathrm{d}x\mathrm{d}t+{\left({\int }_t^{t+\Delta t}\mathbf{f}\mathrm{d}t\right)}_{i+1/2}-{\left({\int }_t^{t+\Delta t}\mathbf{f}\mathrm{d}t\right)}_{i-1/2}={\int }_t^{t+\Delta t}{\int }_{x_{i-1/2}}^{x_{i+1/2}}\mathbf{s}\mathrm{d}x\mathrm{d}t$$ (3)

after removing the source term, for hyperbolic case it is:

$${\int }_t^{t+\Delta t}{\int }_{x_{i-1/2}}^{x_{i+1/2}}\frac{\partial \mathbf{q}}{\partial t}\mathrm{d}x\mathrm{d}t+{\left({\int }_t^{t+\Delta t}\mathbf{f}\mathrm{d}t\right)}_{i+1/2}-{\left({\int }_t^{t+\Delta t}\mathbf{f}\mathrm{d}t\right)}_{i-1/2}=0.$$ (4)

To solve the above equation, the FV approach is used, for a uniform mesh with rectangular grids in the cartesian plane, a conservative form used to update the vector $q$ corresponding to the Eq. (2) is written as:

$$q_i^{n+1}=q_i^n-\frac{\Delta t}{\Delta x}(F_{i+1/2}-F_{i-1/2}),$$ (5)

where

$$q_i^n\approx \frac{1}{\Delta x}{\int }_{x_{i-1/2}}^{x_{i+1/2}}q(x,t^n)dx$$ (6)

is the cell average corresponding to the spatial components, and $F_{i\pm 1/2}$ are the computational flux functions, expressed as the cell averages corresponding to the time component in the form written as:

$$F_{i+1/2}\approx \frac{1}{\Delta t}{\int }_{t^n}^{t^{n+1}}f\left(q(x_{i+1/2},t)\right)dt.$$ (7)

The treatment phase starts at time level T with the cell-averaged values of conserved variables, $q_i^n$. The important task in approximating such conservation laws is the proper selection of the numerical flux presented in the Eq. (7). The numerical flux needs an evaluation with the help of the cell averages, i.e.,

$$F_{i+1/2}=F_{i+1/2}(q_i,q_{i+1}).$$ (8)

On a generalized note, the sort of the construction strategies discussed above demands the solutions of various Riemann problems at each cell interface. Therefore, a comparatively less complex approach is used such as the high resolution TVD approach.

Flux limiting high resolution TVD schemes

A piece-wise linear reconstruction is performed inside each cell for second-order space accuracy (Fig. 2). To avoid spurious oscillations near discontinuities, linear slopes originating from the reconstructed solution should be constrained. It is possible to boost spatial accuracy to second-order by applying the monotone upwind centered schemes for conservation laws (MUSCL) [32], which requires a linear extrapolation, now the numerical flux function is:

$$F_{i+1/2}=F_{i+1/2}(q_{i+1/2}^l,q_{i+1/2}^r),$$ (9)

where $q_{i+1/2}^l$ is a resultant of the linear extrapolation of $q_i$ and $q_{i-1}$ to the left-side of the computational cell $x_{i+1/2}$ (as shown in the Fig. 2), while $q_{i+1/2}^r$ is formed by linearly extrapolating $q_{i+1}$ and $q_{i+2}$ to the right-side of the computational cell $x_{i+1/2}$. But as the process of extrapolation leads to spurious profiles near a shock, because a TVD method is not yielded out of some simple extrapolation process. Hence, the slope-limiting methods are used with slopes denoted as $s^{-}\left(q_i\right)$ and $s^{+}\left(q_{i+1}\right)$. The associated slope at an extremum is placed equal to zero, i.e. first-order. The following extrapolated values are used for the slope limited MUSCL methodology with the minimum-modulus limiter:

$$q_{i+1/2}^l=q_i+\frac{\Delta x}{2}\text{minmod}(s^{-}\left(q_i\right),s^{+}\left(q_i\right)),$$ (10)

$$q_{i+1/2}^r=q_{i+1}-\frac{\Delta x}{2}\text{minmod}(s^{-}\left(q_{i+1}\right),s^{+}\left(q_{i+1}\right)),$$ (11)

where the minmod (short for minimum-modulus) operator is defined as:

$$\text{minmod}(\alpha ,\beta )=\{\begin{array}{cc}\alpha ,\hfill & \left|\alpha \right|\le \left|\beta \right|\text{and}\alpha \beta >0\hfill \\ \beta ,\hfill & \left|\beta \right|\le \left|\alpha \right|\text{and}\alpha \beta >0\hfill \\ 0,\hfill & \alpha \beta \le 0.\hfill \end{array}$$ (12)

The Lax-Friedrichs flux function for the slope limited MUSCL approach [32], is presented as:

$$F_{i+1/2}=\frac{1}{2}(F(q_{i+1/2}^l+F\left(q_{i+1/2}^r\right))-\frac{\Delta x}{2\Delta t}(q_{i+1/2}^r-q_{i+1/2}^l).$$ (13)

In a high resolution method, the numerical flux function is calculated by blending together the numerical fluxes of complementary orders. For $f\left(q\right)=aq$, with a positive speed ‘a’, the formulation for the Lax-Wendroff numerical method is:

$$q_i^{n+1}=q_i^n-\nu (q_i^n-q_{i-1}^n)-\frac{1}{2}\nu (1-\nu )(q_{i+1}^n-2q_i^n+q_{i-1}^n),$$ (14)

where $a.\Delta t/\Delta x$ is known as the Courant number, which is represented by $\nu$. This scheme is a first order upwind technique along with an anti-diffusive component of second order. The numerical method in the Eq. (14) is second-order accurate, but it still does not follows the TVD property. Therefore, the Eq. (14) is further modified by introducing a limiter function, say, $\varphi$ in the following manner:

$$q_i^{n+1}=q_i^n-(q_i^n-q_{i-1}^n)[\nu +\frac{1}{2}\nu (1-\nu )(\frac{\varphi \left(r_{i+1/2}\right)}{r_{i+1/2}}-\varphi \left(r_{i-1/2}\right)\left)\right],$$ (15)

where the function $r_{i+1/2}$ is defined as:

$$r_{i+1/2}=\frac{q_i^n-q_{i-1}^n}{q_{i+1}^n-q_i^n}.$$ (16)

A high resolution technique is developed whenever the limiting functional component given in the Eq. (15) is positive [35]. To advance the solution in time, the numerical fluxes are computed as:

$$F\left(q_{i+1/2}\right)=f_{i+1/2}^l-\varphi \left(r_i\right)(f_{i+1/2}^l-f_{i+1/2}^h),$$ (17)

in above equation, $f^l$ is representing the low-resolution and $f^h$ is showing the high-resolution [35] flux (numerical) functions. An enhancement to the approach framed on the total variation non increasing theory could be applied to remove or minimise the harmful unrealistic oscillations in the high resolution FVM techniques. The concept of TVD schemes was introduced by Harten [26]. For several systems, the TVD formulations constraint that the overall variation does not increase in time in general, that is for Eq. (1),

$$\sum _{i=1}^{N+1}|q_{i+1}^{n+1}-q_i^{n+1}|\le \sum _{i=1}^{N+1}|q_{i+1}^n-q_i^n|.$$ (18)

The right hand side at the problem is computed by implementing the second-order Runge-Kutta technique at each stage to accommodate the proper boundary treatment. For the implementation of flux limiters [9] in the numerical scheme, the step of reconstructing must itself satisfy the TVD constraint given as:

$$\varphi \left(r\right)=\max \{0,\min \{2r,2\left\}\right\}.$$ (19)

Several frequently employed flux limiting TVD functions could be presented as follows:

• •Minmod limiter: $\max (0,\min (r,1\left)\right)$, Roe (1986) [3].
• •Superbee limiter: $\max (0,\min (2r,1),\min (r,2\left)\right)$, Roe (1986) [30].
• •leer Albada limiter: $\frac{r(r+1)}{r^2+1}$, Leer Albada (1982) [8].
• •MC limiter: $\max (0,\min (2r,(1+r)/2,2\left)\right)$, Leer (1977) [36].
• •leer Leer limiter: $\frac{r+\left|r\right|}{1+\left|r\right|}$; Leer (1974) [9].

Fig. 2.

A sketch demonstrating the piecewise linear reconstruction while performing the FV approximation.

Refer to the citations provided with each limiter for a more complete explanation. Further a pictorial depiction of the above mentioned limiters [37] is shown in the Fig. 3. The work is focused on a core principle of modifying and optimising standard limiters in order to enhance the overall computational results.

Fig. 3.

Graphical view of some classical non-linear flux limiters (top), and some linear flux limiters (bottom) inside the TVD regions.

The new flux-limiter scheme

The current work entails optimising a particular classical flux limiter in order to provide a superior hybrid option. Many flux limiters are available in the literature to prevent discontinuities [34]. Some key notions from the field of fuzzy mathematics are also crucial to optimize the classical techniques.

Preliminaries

Fuzzy sets Fundamentally, a fuzzy set is a crisp set (classical set) with a unique feature that permits every element of the universe of discourse to be associated with this classical set by an appropriate intensity level (called membership value). The frequency of membership relies on the scale of compatibility of a specific entity with the classical set. In fuzzy sets, the mostly employed set for membership values is [0,1]. However, for a crisp set this set restricts to the discrete values $\{0,1\}$. Mathematically, for a classical set T in the universe of discourse U, a fuzzy set $A$ can be written as: $A=\left\{\right(x,\mu \left(x\right))\mid x\in T\},$ where the membership function $\mu$ relates the elements of the classical set T to [0,1]. (Refer to the Fig. 4)

Fig. 4.

Examples of fuzzy sets: singleton set (left), triangular set (center), trapezoidal set (right).

Fuzzy Linguistics The fuzzy logic variables which are made up of a set of words in the sense of Fuzzy mathematics, are known as fuzzy linguistics or simply linguistic terms. These variables assist in connecting the universal set’s entities with a releleert membership value, which can then be used to build a relationship between that entity and the fuzzy set in concern [38]. Fuzzy values are significantly more adaptable to real-life models than crisp values because they capture measurement as a result of baseline data. Fuzzy Hedges/Modifiers Fuzzy modifiers are a key feature of the new limiter’s construction. By altering the membership level for the associated fuzzy sets, fuzzy hedges alter the perception of existing data. Corresponding to the fuzzy set $A$, as mentioned above, some of the frequently used hedges are: the Dilation hedge ($\{(y,\sqrt[p]{\left(\mu \right(y\left)\right)})\mid y\in U\}$), and the Concentration hedge ($\left\{(y,{\left(\mu \left(y\right)\right)}^p)\right)\mid y\in U\}$) [30], here $p$ is a real number. (Refer to the Fig. 5)

Fig. 5.

Examples of Fuzzy modifier functions: Dilation modifier (left), Contrast modifier (center), and Concentration modifier (right).

By optimising the classical limiters, the next portion of this section comprises of establishing some new limiters. In this section, the whole framework is focused on appropriate tuning of different parameters in the flux limiting techniques leveraging suitable fuzzy modifier operators.

Formulation of the new hybrid flux limiter

With the aid of fuzzy modifiers, this section emphasises parameter tuning. The applications of the standard dilation, contrast, and concentration operators are considered for optimising flux limited techniques. This work focuses only on the MC limiter and min-mod limiter, however if it serves the problem, such operators can be enforced on any flux limiter.

This paper focuses on an optimization of the classic MC limiter. In the piece-wise form, the MC limiter can be written as:

$$\varphi \left(r\right)=\{\begin{array}{cc}0,\hfill & r\le 0\hfill \\ 2r,\hfill & 0<r\le 0.\overline{3}\hfill \\ 0.5(1+r),\hfill & 0.\overline{3}<r\le 3\hfill \\ 2,\hfill & \text{else}.\hfill \end{array}$$ (20)

This is optimised by implementing the concentration modifying operator of intensity $p=6$ and $p=8$ to the smooth parts and extreme regions (regions having extrema) respectively, and remaining parts are same [32]. So, the new limiting scheme could be mathematically presented as follows:

$$\varphi \left(r\right)=\max (0,\min (\frac{\frac{2}{3}{\left(\frac{9-3r}{8}\right)}^6+2\frac{3r-1}{8}}{{\left(\frac{9-3r}{8}\right)}^6+\frac{3r-1}{8}},2,\frac{\frac{2}{3}{\left(3r\right)}^6}{(1-3r)+{\left(3r\right)}^6}\left)\right).$$ (21)

The new fuzzified limiter is presented in the Fig. 6. In the framework of Fuzzy Mathematics, this technique brings up an incredible number of flux limiting function possibilities [2]. One more hybrid flux construction could be made by using the concentration modifier function of intensity $p=6$ to the extreme points and the dilation modifier of intensity $p=8$ to sharp areas, as shown in Fig. 6. The Shallow water problem, expressed as the Eq. (2), is computed in the next part to show the efficiency of the proposed fuzzy limiter described in the Eq. (16).

Fig. 6.

The graphical representation of the new fuzzified hybrid fuzzy flux limiters.

Numerical simulation and results

To show the efficiency, robustness, and accuracy of the new scheme, computational results are compared with existing experimental data. It must be observed that, some test cases with analytical solutions are of literary interest only, but still they allow for conclusive assessment of the performance of the computational methods [39]. For the comparison of numerical computations some of the classical flux limiting functions namely the MC limiter, and the MM limiter are used in this section.

Throughout this section of numerical validation [27], the gravitational constant is taken as $g=9.81$ (also known as the acceleration due to gravity) and the standard SI measuring units corresponding to the physical quantities [27], [40] (like m (meters), kg (kilograms), s (seconds), etc.) are ignored in the test cases. Uniform cartesian grids have been used for the space discretization.

To analyse the order of accuracy for the hybrid method proposed in this paper, assume $q(x_i,t^n)$ to be the exact solution, and $q_i^n$ to be the computationally obtained solution corresponding to the $i$th grid value at the $n$th step in time, then the $L_1$ error norm (represented by $\parallel e_n{\parallel }_1$) is given as:

$$\parallel e_n{\parallel }_1=\sum _{i=1}^N|q(x_i,t^n)-q_i^n|\Delta x.$$ (22)

and the $L_{\infty }$ error ($\parallel e_n{\parallel }_{\infty }$) is given as:

$$\parallel e_n{\parallel }_{\infty }=\underset{1\le i\le N}{\max }|q(x_i,t^n)-q_i^n|.$$ (23)

where $N$ represents the computational points.

Numerical wave computation using the idealized dam break model

Dam break problems act as a crucial case study to analyse the capacity of a computational model to address upstream and downstream wave propagations involving changes and shocks in the SWEs. The proposed solution scheme has been assessed by using the dam break model in a rectangular channel with flat topography. The numerical domain is taken as $[-1,1]$, and the step size of the discretized interval is $\Delta x=0.005$. The results for the both the new scheme and the classical scheme have been plotted for $N=400$ grid points. In this work, the initial data used by Tseng et al. in their research article has been used [39]. For computational purposes, MATLAB 2015b version has been used with macOS Mojave, RAM 8 GB and 2.3 GHz Intel Core i5.

Test problems (TPs)

TP 1: Riemann profile in velocity

The form of the initial condition used in this TP for numerically computing the dam-break [27] problem is:

$$h(x,0)=1,$$ (24)

$$u(x,0)=\{\begin{array}{cc}{u}_l=1\hfill & x<0\hfill \\ u_r=-1\hfill & x\ge 0.\hfill \end{array}$$ (25)

For the error analysis related to this test case, refer to the Table 1. Also, the Figs. 7–8 validate that the computational results obtained by using the classical limiting technique and the new limiting technique are comparable. The results are better and improved for the proposed new limiter, refer to the Table 1.

Table 1.

Error analysis for the TP 1 using $L_1$ and $L_{\infty }$ error norms.

$N$	$L_1$ error (classical)	$L_1$ error (new)	$L_{\infty }$ error (classical)	$L_{\infty }$ error (new)
100	4.96e-03	4.45e-03	1.42e-01	1.36e-01
150	3.18e-03	2.27e-03	1.16e-01	1.07e-01
200	2.63e-03	2.39e-03	1.55e-01	1.52e-01
250	2.04e-03	1.84e-03	1.48e-01	1.44e-01

Fig. 7.

Approximation results corresponding to the velocity component for the TP 1 using the proposed scheme and the classic MC scheme for $\Delta t=0.003$ and $N=400$ at $t=0.2$.

Fig. 8.

Computational solution profiles for the TP 1 using the proposed limiting scheme and the classical MC scheme for $\Delta t=0.003$, and $N=400$ at $t=0.2$.

TP 2: Riemann profile in height

For computing the dam-break problem, the form of the initial condition used in this TP is:

$$u(x,0)=0,$$ (26)

$$h(x,0)=\{\begin{array}{cc}{h}_l=0.01\hfill & x<0\hfill \\ h_r=2\hfill & x\ge 0.\hfill \end{array}$$ (27)

Figures 9-10 show that the numerical outputs achieved by the MC limiter and the hybrid limiter are comparable. Although, in the Fig. 10, the solution from the new limiter dips at about $x=-0.2$, while the original version seems to capture the solution better. This is the point of transcritical flow, where linearised Riemann solvers usually require an entropy fix. Still, the new method is capable of generating a nice comparable resolution, with smooth solutions.

Fig. 9.

Approximation results corresponding to the velocity component for the TP 2 using the proposed limiting scheme and the classical MC scheme for $\Delta t=0.003$ at $t=0.2$ and $N=400$.

Fig. 10.

Approximation results corresponding to the height component for the TP 2 using the proposed limiting scheme and the classical MC scheme for $\Delta t=0.003$ and $N=400$ at $t=0.2$.

In the Table 2 the point-wise errors are presented. The improvised trait of the new method as compared to the classical method is validated by the numerical computations and the error magnitude comparison.

Table 2.

Error analysis for the TP 2 using $L_1$ and $L_{\infty }$ error norms.

$N$	$L_1$ error (classical)	$L_1$ error (new)	$L_{\infty }$ error (classical)	$L_{\infty }$ error (new)
100	5.64e-03	4.77e-03	2.04e-01	9.15e-02
150	2.62e-03	2.39e-03	1.32e-01	7.35e-02
200	2.16e-03	2.01e-03	1.66e-01	1.05e-01
250	1.96e-03	1.62e-03	1.97e-01	1.30e-01

TP 3: Riemann profile in velocity

The form of the initial condition used in this TP for numerically computing the dam-break [27] problem is:

$$h(x,0)=1,$$ (28)

$$u(x,0)=\{\begin{array}{cc}{u}_l=-1\hfill & x<0\hfill \\ u_r=1\hfill & x\ge 0.\hfill \end{array}$$ (29)

Figure 11 validates that the approximated results are giving better convergence using the new optimised limiting scheme. A nice improvisation could be observed in the solution profiles. The outputs of both the depth and height solutions are comparable [1].

Fig. 11.

Approximation results corresponding to the TP 3 for $N=400$, obtained by imposing the new and the classical scheme for $\Delta t=0.003$ at the time $t=0.2$. Here velocity profile ($u$) is on the right, and the height profile ($h$) is on the left.

TP 4: Riemann profile in height

The form of the initial condition used in this TP for numerically computing the dam-break [27] problem is:

$$u(x,0)=0,$$ (30)

$$h(x,0)=\{\begin{array}{cc}{h}_l=2\hfill & x<0\hfill \\ h_r=0.01\hfill & x\ge 0.\hfill \end{array}$$ (31)

Table 4 shows the point-wise flaws caused by the classical technique and the new technique for the $L_1$ norm at $t=0.2$ units, and the Fig. 12 shows the approximate solution profile obtained by employing the classic MC limiting scheme and the new one. It is observable that the results are comparable. In fact, the computational output through the new limiter appears to be slightly improved, as shown in the Table 4, with the exception of slight dissipation at the corners.

Table 4.

Error analysis for the TP 4 using the $L_1$ and $L\infty$ error norms.

$N$	$L_1$ error (classical)	$L_1$ error (new)	$L_{\infty }$ error (classical)	$L_{\infty }$ error (new)
100	4.98e-03	4.32e-03	3.16e-01	7.51e-02
150	3.07e-03	2.95e-03	1.56e-01	5.17e-02
200	2.54e-03	1.97e-03	1.43e-01	1.12e-01
250	1.63e-03	1.02e-03	1.27e-01	1.18e-01

Fig. 12.

Approximation results for the height profile corresponding to the TP 4 using the proposed limiter and the MC limiter for $\Delta t=0.003$ at $t=0.2$ and $N=400$.

Table 4 gives the error analysis for the classical and the proposed scheme for the $L_1$ and the $L_{\infty }$ norms. It is clear from the output data, that the MC limiter is capable of capturing the solution profile with a slight oscillation profile near $x=0.5$. However, the numerical outcome obtained from the new limiter looks more promising.

Notations

The following is the list of all the notations/symbols used in this work:

• •x: space variable
• •t: time variable
• •q: vector of flow components
• •u: flow velocity component
• •h: height component
• •g: gravitational constant
• •N: number of computational points
• •f: physical flux function
• •s: source term
• •$\eta$: manning constant
• •$\alpha$, $\beta$: some arbitrary values
• •Z: topography function
• •i: sub-script for spatial discretization
• •n: super-script for temporal integration
• •$\Delta x$: spatial step size
• •$\Delta t$: temporal step
• •F: flux function (numerical)
• •$q^l$: linear extrapolation corresponding to the left side of cell interface
• •$q^r$: linear extrapolation corresponding to the right side of cell interface
• •$s^{-}$: slope function at left interface
• •$s^{+}$: slope function at right interface
• •$\nu$: Courant number (CFL number)
• •$\varphi$: limiter function
• •$f^l$: low order numerical flux
• •$f^h$: high order numerical flux
• •$\mu$: membership function
• •T: classical/crisp set
• •p: any real number
• •U: Universal set
• •y: an element from the universal set U
• •$e_n$: error term at $n$th time step
• •s: seconds
• •kg: kilograms
• •m: meters

Contributions

All authors contributed equally to the planning, execution, and analysis of this research paper.

Funding

This work was not supported by any organization.

Discussion and remarks

The study has presented the numerical approximation of one-dimensional SWEs with dam break flow problem. Here, the novelty of this research lies in the assessment of the performance of a new fuzzy logic based flux limiting scheme for the shallow water model.

• •This concept offers a new way to compute the numerical results of shallow water flows with the aid of a completely different subject: Fuzzy Mathematics.
• •The basic approach to tackle the problem is based as usual on the rules of conservation.
• •The major difference between the classical and the new scheme lies in its important key feature namely the Fuzzy modifiers.
• •On smooth solutions, the method has the second order of convergence, while on discontinuities, it has the first order of convergence, which corresponds to the accuracy of Godunov-type techniques.
• •From the experiments it has revealed clearly that the optimized scheme is more accurate than the classical schemes because the error term in the new scheme are less than the MM and MC scheme, as seen in the Table 1, Table 2, Table 3, Table 4.

Table 3.

Error analysis for the TP 3 using $L_1$ and $L_{\infty }$ error norms.

$N$	$L_1$ error (classical)	$L_1$ error (new)	$L_{\infty }$ error (classical)	$L_{\infty }$ error (new)
100	2.13e-03	1.97e-03	4.00e-02	3.86e-02
150	1.24e-03	1.12e-03	2.84e-02	2.66e-02
200	8.68e-04	7.75e-04	2.77e-02	2.51e-02
250	6.27e-04	5.50e-04	2.19e-02	1.97e-02

In this work, we have proposed the one dimensional shallow water model with the fuzzy FV method. Similar work may be extended to higher dimensions, and many real-life applications such as safety analysis in case of flood events etc.; we leave these ideas as our future work. The results demonstrate the applicability and benefits of indulging Fuzzy logic to form the numerical techniques.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

• No data was used for the research described in the article.