Fractional calculus and the ESR test

J. Vanterler da C. Sousa; E. Capelas de Oliveira; L. A. Magna; J. Vanterler da C. Sousa; E. Capelas de Oliveira; L. A. Magna

doi:10.3934/Math.2017.4.692

AIMS Mathematics

2017, Volume 2, Issue 4: 692-705. doi: 10.3934/Math.2017.4.692

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Research article

Fractional calculus and the ESR test

1.
Department of Applied Mathematics, Imecc–Unicamp, Sérgio Buarque de Holanda, 651 13083–859, Campinas, SP, Brazil
2.
Department of Medical Genetics, School of Medical Sciences–Unicamp, 13083–887, Campinas, SP, Brazil

Received: 21 August 2017 Accepted: 08 December 2017 Published: 15 December 2017
MSC : 26A33, 33RXX, 34A30, 35KXX, 92BXX

We consider the partial differential equation of a mathematical model proposed by Sharma et al. ^[1] to describe the concentration of nutrients in blood, a factor which influences erythrocyte sedimentation rate. Introducing in it a fractional derivative in the Caputo sense, we create a new, timefractional mathematical model which contains, as a particular case, the original model. We obtain an analytic solution of this time-fractional partial differential equation in terms of Mittag-Leffler and Wright functions and to show that our model is more realistic than the Sharma model.

Keywords:

Citation: J. Vanterler da C. Sousa, E. Capelas de Oliveira, L. A. Magna. Fractional calculus and the ESR test[J]. AIMS Mathematics, 2017, 2(4): 692-705. doi: 10.3934/Math.2017.4.692

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Abstract

1. Introduction

Hydraulic jumps are frequently observed in lab studies, river flows, and coastal environments. The shallow water equations, commonly known as Saint-Venant equations, have important applications in oceanography and hydraulics. These equations are used to represent fluid flow when the depth of the fluid is minor in comparison to the horizontal scale of the flow field fluctuations ^[1]. They have been used to simulate flow at the atmospheric and ocean scales, and they have been used to forecast tsunamis, storm surges, and flow around constructions, among other things ^[2,3]. Shallow water is a thin layer of constant density fluid in hydrostatic equilibrium that is bordered on the bottom by a rigid surface and on the top by a free surface ^[4].

Nonlinear waves have recently acquired importance due to their capacity to highlight several complicated phenomena in nonlinear research with spirited applications ^[5,6,7,8]. Teshukov in ^[9] developed the equation system that describes multi-dimensional shear shallow water (SSW) flows. Some novel solitary wave solutions for the ill-posed Boussinesq dynamic wave model under shallow water beneath gravity were constructed ^[10]. The effects of vertical shear, which are disregarded in the traditional shallow water model, are included in this system to approximate shallow water flows. It is a non-linear hyperbolic partial differential equations (PDE) system with non-conservative products. Shocks, rarefactions, shear, and contact waves are all allowed in this model. The SSW model can capture the oscillatory character of turbulent hydraulic leaps, which corrects the conventional non-linear shallow water equations' inability to represent such occurrences ^[11]. Analytical and numerical techniques for solving such problems have recently advanced; for details, see ^{[12,13,14,15,16,17]} and the references therein.

The SSW model consists of six non-conservative hyperbolic equations, which makes its numerical solution challenging since the concept of a weak solution necessitates selecting a path that is often unknown. The physical regularization mechanism determines the suitable path, and even if one knows the correct path, it is challenging to construct a numerical scheme that converges to the weak solution since the solution is susceptible to numerical viscosity ^[18].

In the present work, we constructed the generalized Rusanov (G. Rusanov) scheme to solve the SSW model in 1D of space. This technique is divided into steps for predictors and correctors ^{[19,20,21,22,23]}. The first one includes a numerical diffusion control parameter. In the second stage, the balance conservation equation is retrieved. Riemann solutions were employed to determine the numerical flow in the majority of the typical schemes. Unlike previous schemes, the interesting feature of the G. Rusanov scheme is that it can evaluate the numerical flow even when the Riemann solution is not present, which is a very fascinating advantage. In actuality, this approach may be applied as a box solver for a variety of non-conservative law models.

The remainder of the article is structured as follows. Section 2 offers the non-conservative shear shallow water model. Section 3 presents the structure of the G. Rusanov scheme to solve the 1D SSW model. Section 4 shows that the G. Rusanov satisfies the C-property. Section 5 provides several test cases to show the validation of the G. Rusanov scheme versus Rusanov, Lax-Friedrichs, and reference solutions. Section 6 summarizes the work and offers some conclusions.

2. Mathematical formulation for the shear shallow water model

The one-dimensional SSW model without a source term is

$\begin{equation} \frac{\partial \mathrm{W}}{\partial t} + \frac{\partial \mathit{F}(\mathrm{W})}{\partial x}+\mathrm{K( \mathrm{W})}\frac{\partial \mathrm{h}}{\partial x} = \mathrm{S(\mathrm{W})}, \end{equation}$

(2.1)

$\begin{equation*} \label{equa3} \mathrm{W} = \left(\begin{array}{c} \mathrm{h} \\ \mathrm{h v_1}\\ \mathrm{h v_2}\\ \mathrm{E_{11}}\\ \mathrm{E_{12}}\\ \mathrm{E_{22}}\\ \end{array}\right),\qquad {\mathit{F}(\mathrm{W})} = \left(\begin{array}{c} \mathrm{h v_1} \\ \mathrm{R_{11}}+\mathrm{h v^2_{1}}+g\frac{\mathrm{h^2}}{2} \\ \mathrm{R_{12}}+\mathrm{h v_{1}v_2} \\ (\mathrm{E_{11}}+\mathrm{R_{11}})\mathrm{v_1} \\ \mathrm{E_{12} v_1}+\frac{1}{2}(\mathrm{R_{11}v_2}+\mathrm{R_{12}}v_1) \\ \mathrm{E_{22} v_1}+\mathrm{R_{12}v_2} \\ \end{array}\right), \end{equation*}$

$\begin{equation*} \label{equa3k} \mathrm{K(\mathrm{W})} = \left(\begin{array}{c} 0 \\ 0\\ 0\\ g\mathrm{h v_1}\\ \frac{1}{2}g\mathrm{hv_2}\\ 0\\ \end{array}\right), \mathrm{S(\mathrm{W})} = \left(\begin{array}{c} 0 \\ -gh \frac{\partial \mathrm{B}}{\partial x}- C_f|v|v_1\\ -C_f|v|v_2\\ -ghv_1 \frac{\partial \mathrm{B}}{\partial x}+\frac{1}{2} D_{11} -C_f|v|v_{1}^2\\\\ -\frac{1}{2} ghv_2 \frac{\partial \mathrm{B}}{\partial x}+\frac{1}{2} D_{12}-C_f|v|v_1v_2\\\\ \frac{1}{2} D_{22}-C_f|v|v_{2}^2\\ \end{array}\right), \end{equation*}$

while $\mathrm{R_{11}}$ and $\mathrm{R_{12}}$ are defined by the following tensor $\mathrm{R_{ij}} = \mathrm{h p_{i j}}$ , and also, $\mathrm{E_{11}}$ , $\mathrm{E_{12}}$ and $\mathrm{E_{22}}$ are defined by the following tensor $\mathrm{E_{i j}} = \frac{1}{2}\mathrm{R_{ij}}+\frac{1}{2}\mathrm{\mathrm{h v_i v_j}}$ with $i\ge 1, j\le 2$ . We write the previous system 2.1 in nonconservative form with non-dissipative $(C_f = 0, D_{11} = D_{12} = D{22} = 0)$ , see ^[24], which can be written as follows

$\begin{equation} \frac{\partial \mathrm{W}}{\partial t} + (\nabla{\mathit{F}(\mathrm{W})} +\mathit{C(\mathrm{W}})) \frac{\partial \mathrm{W}}{\partial x} = \mathrm{S_1(\mathrm{W})}. \end{equation}$

(2.2)

Also, we can write the previous system as the following

$\begin{equation} \frac{\partial \mathrm{W}}{\partial t} + \mathit{A(\mathrm{W})} \frac{\partial \mathrm{W}}{\partial x} = \mathrm{S_1(\mathrm{W})} \end{equation}$

(2.3)

with

$\begin{equation*} \mathit{A(\mathrm{W})} = \begin{bmatrix} 0 & 1 & 0 & 0 &0 & 0 \\ \mathrm{gh} & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 \\ -3\frac{\mathrm{E_{11} v_1}}{\mathrm{h}}+2v^{3}_1 + \mathrm{g h v_2} &3\frac{\mathrm{E_{11}}}{\mathrm{h}}-3v^{2}_1 & 0 & 3 \mathrm{v_1} & 0 & 0 \\ -2\frac{\mathrm{E_{12} v_1}}{\mathrm{h}}-\frac{\mathrm{E_{11}v_2}}{\mathrm{h}}+ \mathrm{2v^{2}_1 v_2}+\frac{\mathrm{g h v_2}}{2} & 2\frac{\mathrm{E_{12}}}{h}-2\mathrm{v_1 v_2} & \frac{\mathrm{E_{11}}}{h}- \mathrm{v^{2}_1} & \mathrm{v_2} & 2\mathrm{v_1} & 0 \\ -\frac{\mathrm{E_{22} v_1}}{\mathrm{h}}-2\frac{\mathrm{E_{12}v_2}}{\mathrm{h}}+ \mathrm{v^{2}_2 v_1}& \frac{\mathrm{E_{22}}}{\mathrm{h}}+ \mathrm{v^{2}_2}& 2\frac{\mathrm{E_{12}}}{\mathrm{h}}-2\mathrm{v_2 v_1}& 0 &2\mathrm{v_2}& \mathrm{v_1} \\ \end{bmatrix} \end{equation*}$

and

$\begin{equation*} \label{equa3km} \mathrm{S_1(\mathrm{W})} = \left(\begin{array}{c} 0 \\ -gh \frac{\partial \mathrm{B}}{\partial x}\\ 0\\ -ghv_1 \frac{\partial \mathrm{B}}{\partial x}\\\\ -\frac{1}{2} ghv_2 \frac{\partial \mathrm{B}}{\partial x}\\\\ 0\\ \end{array}\right). \end{equation*}$

System (2.3) is a hyperbolic system and has the following eigenvalues

$\lambda_1 = \lambda_2 = v_1, \quad \lambda_3 = v_1-\sqrt{2\frac{\mathrm{E_{11}}}{h}-v^2_1}, \quad \lambda_4 = v_1+\sqrt{2\frac{\mathrm{E_{11}}}{h}-v^2_1},$

$\lambda_5 = v_1-\sqrt{\mathrm{g h}+3(v^2_1-2\frac{\mathrm{E_{11}}}{h})}, \quad \lambda_6 = v_1+\sqrt{\mathrm{g h}+3(v^2_1-2\frac{\mathrm{E_{11}}}{h})}.$

3. The G. Rusanov scheme for SSWE

In order deduce the G. Rusanov scheme, we rewrite the system (2.1) as follows

$\begin{equation} \frac{\partial \mathrm{W}}{\partial t} + \frac{\partial \mathit{F}(\mathrm{W})}{\partial x} = -\mathrm{K( \mathrm{W})}\frac{\partial \mathrm{h}}{\partial x}+\mathrm{S_1(\mathrm{W})} = \mathrm{Q(\mathrm{W})}. \end{equation}$

(3.1)

Integrating Eq (3.1) over the domain $[t_n, t_{n+1}]\times[x_{i-\frac{1}{2}}, x_{i+\frac{1}{2}}]$ gives the finite-volume scheme

$\begin{equation} \mathrm{W_i^{n+1}} = \mathrm{W_i^n} - \frac{\Delta{t}}{\Delta{x}}\left(\mathit{F}\left(\mathrm{W^n_{i+1/2}}\right) - \mathit{F}\left(\mathrm{W^n_{i-1/2}}\right)\right)+\Delta t {\mathrm{Q_{i}}}^{n} \end{equation}$

(3.2)

in the interval $[x_{i-1/2}, x_{i+1/2}]$ at time $t_n$ . $\mathrm{W_i^n}$ represents the average value of the solution $\mathrm{W}$ as follows

$\mathrm{W_i^n} = \frac{1}{\Delta{x}}\int^{x_{i+1/2}}_{x_{i-1/2}}\mathrm{W}(t_{n},x)\;dx,$

and $\mathit{F}(\mathrm{W_{i\pm1/2}^n)}$ represent the numerical flux at time $t_n$ and space $x = x_{i\pm1/2}$ . It is necessary to solve the Riemann problem at $x_{i+1/2}$ interfaces due to the organization of the numerical fluxes. Assume that for the first scenario given below, there is a self-similar Riemann problem solution associated with Eq (3.1)

$\begin{equation} \mathrm{W(x,0)} = \begin{cases} \mathrm{W_L}, & \text{if}\quad x < 0,\\[2ex] \mathrm{W_R}, & \text{if}\quad x > 0 \end{cases} \end{equation}$

(3.3)

is supplied by

$\begin{equation} \mathrm{W(t,x)} = \mathrm{R_s}\left(\frac xt,\mathrm{W_L},\mathrm{W_R}\right). \end{equation}$

(3.4)

The difficulties with discretization of the source term in (3.2) could arise from singular values of the Riemann solution at the interfaces. In order to overcome these challenges and reconstruct a $\mathrm{W_{i+1/2}^n}$ approximation, we created a finite-volume scheme in ^{[19,20,21,22,23,25,26,27]} for numerical solutions of conservation laws including source terms and without source terms. The principal objective here is building the intermediate states $\mathrm{W_{i\pm1/2}^n}$ to be utilized in the corrector stage (3.2). The following is obtained by integrating Eq (3.1) through a control volume $[t_n, t_{n}+\theta^n_{i+1/2}]\times[x^-, x^+]$ , that includes the point $(t_n, x_{i+1/2})$ , with the objective to accomplish this:

$\begin{eqnarray} \int_{x^-}^{x^+}\mathrm{W(t_{n}+\theta^n_{i+1/2},x)}\;dx & = & \Delta x^-\mathrm{W_i^n} + \Delta x^+\mathrm{W_{i+1}^n} -\\ &&\theta^n_{i+1/2}\Bigl(\mathit{F}(\mathrm{W^n_{i+1}}) \!-\! \mathit{F}(\mathrm{W^n_{i}})\Bigr)\!+\!\theta^n_{i+1/2}(\Delta x^{-}\!-\!\Delta x^+)\mathrm{Q^n_{i\!+\!\frac{1}{2}}} \end{eqnarray}$

(3.5)

with $\mathrm{W_{i\pm1/2}^n}$ denoting the approximation of Riemann solution $\mathrm{R_s}$ across the control volume $[x^-, x^+]$ at time $t_{n}+\theta^n_{i+1/2}$ . While calculating distances $\Delta x^-$ and $\Delta x^+$ as

$\Delta x^- = \left|x^- - x_{i+1/2}\right|,\qquad \Delta x^+ = \left|x^+ - x_{i+1/2}\right|,$

$\mathrm{Q^n_{i+\frac{1}{2}}}$ closely resembles the average source term $\mathrm{Q}$ and is given by the following

$\begin{equation} \mathrm{Q^n_{i+\frac{1}{2}}} = \frac{1}{\theta^n_{i+\frac{1}{2}}(\Delta x^- + \Delta x^+)} \int_{t_n}^{t_n+\theta^n_{i+\frac{1}{2}} }\int_{x^-}^{x^+}{\mathrm{Q}}(\mathrm{W})dt\;dx . \end{equation}$

(3.6)

Selecting $x^- = x_i$ , $x^+ = x_{i+1}$ causes the Eq (3.5) to be reduced to the intermediate state, which given as follows

$\begin{equation} \mathrm {W_{i+1/2}^{n}} = \frac12\left(\mathrm{W_i^n} + \mathrm{W_{i+1}^n}\right) - \frac{\theta^n_{i+1/2}}{\Delta{x}}\Bigl(\mathit{F}(\mathrm{W^n_{i+1}}) - \mathit{F}(\mathrm{W^n_{i}})\Bigr)+\theta^n_{i+\frac{1}{2}}\mathrm{Q^n_{i+\frac{1}{2}}}, \end{equation}$

(3.7)

where the approximate average value of the solution $\mathrm{W}$ in the control domain $[t_n, t_{n}+\theta^n_{i+1/2}]\times[x_{i}, x_{i+1}]$ is expressed by $\mathrm{W_{i+1/2}^n}$ as the following

$\begin{equation} \mathrm{{W}_{i+1/2}^n} = \frac{1}{\Delta x}\int_{x_i}^{x_{i+1}}{W}(x,t_n+\theta^n_{i+1/2})dx \end{equation}$

(3.8)

by selecting $\theta^n_{i+1/2}$ as follows

$\begin{equation} \theta^n_{i+1/2} = {\alpha^n_{i+1/2}}\bar{\theta}_{i+1/2}\quad ,\qquad \bar{\theta}_{i+1/2} = \frac{\Delta x}{2S_{i+1/2}^n}. \end{equation}$

(3.9)

This choice is contingent upon the results of the stability analysis^[19]. The following represents the local Rusanov velocity

$\begin{equation} S_{i+1/2}^n = \underset{k = 1,...,K}{max}(max(\mid \lambda^n_{k,i} \mid,\mid \lambda^n_{k,i+1} \mid )), \end{equation}$

(3.10)

where $\alpha^n_{i+1/2}$ is a positive parameter, and $\lambda^n_{k, i}$ represents the $k^{th}$ eigenvalues in (2.3) evaluated at the solution state $\mathrm{W^n_{i}}$ . Here, in our case, $k = 6$ for the shear shallow water model. One can use the Lax-Wendroff technique again for $\alpha^n_{i+1/2} = \frac{\Delta t}{\Delta x S^n_{i+1/2}}$ . Choosing the slope $\alpha^n_{i+1/2} = \tilde\alpha_{i+1/2}^n$ , the proposed scheme became a first-order scheme, where

$\begin{equation} \tilde\alpha_{i+1/2}^n = \frac{S_{i+1/2}^n}{s_{i+1/2}^n}, \end{equation}$

(3.11)

and

$\begin{equation} s_{i+1/2}^n = \underset{k = 1,...,K}{\min}(\max(\mid \lambda^n_{k,i} \mid,\mid \lambda^n_{k,i+1} \mid )). \end{equation}$

(3.12)

In this case, the control parameter can be written as follows:

$\begin{equation} \mathrm{\alpha_{i+1/2}^n} = \mathrm{\tilde\alpha_{i+1/2}^n} + \sigma_{i+1/2}^n\Phi\left(\mathrm{r^m_{i+1/2}}\right), \end{equation}$

(3.13)

where $\mathrm{\tilde \alpha_{i+1/2}^n}$ is given by (3.11), and $\Phi_{i+1/2} = \Phi\left(r_{i+1/2}\right)$ is a function that limits the slope. While for

$\begin{equation*} \mathrm{r_{i+1/2}} = \frac{\mathrm{W_{i+1-q}}-\mathrm{W_{i-q}}}{\mathrm{W_{i+1}}-\mathrm{W_{i}}},\qquad q = \mathrm{sgn}\left[\mathit{F'}(X^{n+1},\mathrm{W^n_{i+1/2}})\right] \end{equation*}$

and

$\begin{equation*} \sigma_{i+1/2}^n = \frac{\Delta t}{\Delta x}S_{i+1/2}^n - \frac{S_{i+1/2}^n}{s_{i+1/2}^n}. \end{equation*}$

One may use any slope limiter function, including minmod, superbee, and Van leer ^[28] and ^[29]. At the end, the G. Rusanov scheme for Eq (2.3) can be written as follows

$\begin{equation} \left \{ \begin{array}{ll} \mathrm{W}^n_{i+\frac{1}{2}} = \frac{1}{2}( \mathrm{W}^n_i+ \mathrm{W}^n_{i+1})-\frac{\alpha^n_{i+\frac{1}{2}}}{2 S^n_{j+\frac{1}{2}}} \left [\mathit{F}( \mathrm{W}^n_{i+1})-\mathit{F}( \mathrm{W}^n_i) \right ]+\frac{\alpha^n_{i+\frac{1}{2}}}{2 S^n_{j+\frac{1}{2}}} \mathrm{Q^n_{i+\frac{1}{2}}},\\ \mathrm{W}_{i}^{n+1} = \mathrm{W}_{i}^{n}-r^{n}\left[ \mathit{F}\left( \mathrm{W}_{i+{\frac{1}{2}}}^{n}\right) -\mathit{F}\left( \mathrm{W}_{i-{\frac{1}{2}}}^{n}\right) \right]+\Delta t^n {\mathrm{Q_{i}}}^{n}. \end{array} \right. \end{equation}$

(3.14)

4. How the source term is treated

With a definition of the C-property ^[30], the source term in (2.1) is discretized in the G. Rusanov method in a way that is well-balanced with the discretization of the flux gradients. According to ^[30,31], if the following formulas hold, a numerical scheme is said to achieve the C-property for the system (3.1).

$\begin{equation} (hp_{11})^n = \mbox{ constant},\quad \quad h^n+B = \mbox{ constant}\quad \mbox{and}\quad v^n_1 = v^n_2 = 0. \end{equation}$

(4.1)

When we set $v_1 = v_2 = 0$ in the stationary flow at rest, we get system (3.1), which can be expressed as follows.

$\begin{equation} \begin{array}{ll} \frac{\partial}{\partial t} \left(\begin{array}{c} h \\ 0 \\ 0\\ \frac{1}{2}h p_{11}\\ \frac{1}{2}h p_{12}\\ \frac{1}{2}h p_{22}\\ \end{array}\right)+ \frac{\partial}{\partial x} \left(\begin{array}{c} 0 \\ h p_{11}+\frac{1}{2}gh^2\\ h p_{12}\\ 0\\ 0\\ 0\\ \end{array}\right) = \left(\begin{array}{c} 0 \\ -gh\frac{\partial B}{\partial x} \\ 0\\ 0\\ 0\\ 0\\ \end{array}\right) \end{array} = Q(x,t). \end{equation}$

(4.2)

We can express the predictor stage (3.14) as follows after applying the G. Rusanov scheme to the previous system

$\begin{equation} \mathbb{W}^n_{i+\frac{1}{2}} = \left(\begin{array}{c} \frac{1}{2}({h}^n_i+{h}^n_{i+1}) \\ -\frac{\alpha^n_{i+\frac{1}{2}}}{4 S^n_{i+\frac{1}{2}}}({h}^n_i+{h}^n_{i+1})[({h}^n_{i+1}+ B_{i+1}) - ({h}^n_{i}+ B_{i})]- \frac{\alpha^n_{i+\frac{1}{2}}}{2 S^n_{i+\frac{1}{2}}}( (hp_{11})^n_{i+1}- (hp_{11})^n_{i}\\ -\frac{\alpha^n_{i+\frac{1}{2}}}{2 S^n_{i+\frac{1}{2}}} ((hp_{12})^n_{i+1}- (hp_{12})^n_{i} )\\ \frac{1}{4}((hp_{11})^n_{i+1}+ (hp_{11})^n_{i})\\ \frac{1}{4}((hp_{12})^n_{i+1}+(hp_{12})^n_{i})\\ \frac{1}{4}((hp_{22})^n_{i+1}+(hp_{22})^n_{i})\\ \end{array}\right).\\ \end{equation}$

(4.3)

Also, we can write the previous equations as follows

$\begin{equation} \mathrm{W}^n_{i+\frac{1}{2}} = \left(\begin{array}{c} {h}^n_{i+\frac{1}{2}} \\ 0\\ -\frac{\alpha^n_{i+\frac{1}{2}}}{2 S^n_{i+\frac{1}{2}}} ((hp_{12})^n_{i+1}- (hp_{12})^n_{i} )\\ \frac{1}{2}(hp_{11})^n_{i+\frac{1}{2}}\\ \frac{1}{2}(hp_{12})^n_{i+\frac{1}{2}}\\ \frac{1}{2}(hp_{22})^n_{i+\frac{1}{2}}\\ \end{array}\right),\\ \end{equation}$

(4.4)

and the stage of the corrector updates the solution to take on the desired form.

$\begin{equation} \left(\begin{array}{c} (h)^{n+1}_i \\ (h v_1)^{n+1}_i\\ (h v_2)^{n+1}_i \\ (E_{11})^{n+1}_i\\ (E_{12})^{n+1}_i\\ (E_{22})^{n+1}_i \end{array}\right) = \left(\begin{array}{c} (h)^{n}_i \\ (h v_1)^{n}_i\\ (h v_2)^{n}_i \\ (E_{11})^{n}_i\\ (E_{12})^{n}_i\\ (E_{22})^{n}_i\\ \end{array}\right)-\frac{ \Delta t_n}{ \Delta x} \left(\begin{array}{c} 0 \\ \frac{g}{2} ( ({h}^n_{i+\frac{1}{2}})^2-({h}^n_{i-\frac{1}{2}})^2 )\\ (h p_{12})^n_{i+\frac{1}{2}}-(h p_{12})^n_{i-\frac{1}{2}}\\ 0\\ 0\\ 0 \end{array}\right) \end{equation}$

(4.5)

$\begin{equation*} + \left(\begin{array}{c} 0 \\ \Delta t_n Q^n_i\\ 0 \\ 0\\ 0\\ 0 \end{array}\right). \end{equation*}$

The solution is stationary when $\mathrm{W}^{n+1}_i = \mathrm{W}^n_i$ and this leads us to rewrite the previous system (4.5) as follows

$\frac{\Delta t_n}{2 \Delta x }g((h^n_{i+\frac{1}{2}})^2-(h^n_{i-\frac{1}{2}})^2 ) -\Delta t_n Q^n_i.$

This lead to the following

$\begin{equation} Q^n_i = -\frac{g}{8 \Delta x}(h^n_{i+1}+2 h^n_i+h^n_{i-1})(B_{i+1}-B_{i-1}). \end{equation}$

(4.6)

Afterward, when the source term in the corrector stage is discretized in the earlier equations, this leads to the G. Rusanov satisfying the C-property.

5. Numerical simulation for the SSW model

In order to simulate the SSW system numerically, we provide five test cases without a source term using the G. Rusanov, Rusanov, and Lax-Friedrichs schemes to illustrate the effectiveness and precision of the suggested G. Rusanov scheme. The computational domain for all cases is $\mathbb{L} = [0, 1]$ divided into $300$ gridpoints, and the final time is $t = 0.5s$ , except for test case $2$ , where the final time is $t = 10s$ . We evaluate the three schemes with the reference solution on the extremely fine mesh of $30000$ cells that was produced by the traditional Rusanov scheme. Also, we provide one test case with a source term using the G. Rusanov scheme in the domain $[0, 1]$ divided into $200$ gridpoints and final time $t = 0.5$ . We select the stability condition ^[19] in the following sense

$\begin{equation} \mathrm{\Delta t} = \mathrm{CFL}\frac{\mathrm{\Delta x}}{\max\limits_{i}\bigl(\left|\mathrm{\alpha_{i+\frac{1}{2}}^n}\mathrm{S_{i+\frac{1}{2}}^n}\right|\bigr)}, \end{equation}$

(5.1)

where a constant $\mathrm{CFL} = 0.5$ .

5.1. Test case 1

This test case was studied in ^[24] and ^[32], and the initial condition is given by

$\begin{equation} (h, v_1, v_2, p_{11}, p_{22}, p_{12}) = \begin{cases} (0.01,\quad 0.1,\quad 0.2,\quad 0.04,\quad 0.04,\quad 1\times 10^{-8}) \quad \mbox{if} \quad x \le \frac{\mathbb{L}}{2},\\[2ex] (0.02,\quad 0.1,\quad -0.2,\quad 0.04,\quad 0.04,\quad 1\times 10^{-8} ) \quad \mbox{if} \quad x > \frac{\mathbb{L}}{2}.\\ \end{cases} \end{equation}$

(5.2)

The solution include five waves, which are represented by $1-$ shock and $6-$ rarefaction. We compare the numerical results with the reference solution, which is calculated with the Rusanov scheme with very fine mesh of $30000$ cells. Figures 1–3 show the numerical results and variation of parameter of control. We note that all waves are captured by this scheme and the numerical solution agrees with the reference solution. Table 1 shows the CPU time of computation for the three schemes by using a Dell i5 laptop, CPU 2.5 GHZ.

Figure 1. High water

$h$ and velocity

$v_1$ .

Scheme	G. Rusanov	Rusanov	Lax-Friedrichs
CPU time	0.068005	0.056805	0.0496803

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AIMS Mathematics

Fractional calculus and the ESR test

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Abstract

1. Introduction

2. Mathematical formulation for the shear shallow water model

3. The G. Rusanov scheme for SSWE

4. How the source term is treated

5. Numerical simulation for the SSW model

5.1. Test case 1

5.2. Test case 2

5.3. Test case 3

5.4. Test case 4

5.5. Test case 5

5.6. Test case 6 with a source term

6. Conclusions

Author contributions

Use of AI tools declaration

Acknowledgments

Conflict of interest

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