# Multidimensional Convection and Diffusion Problems

(Difference between revisions)
 Revision as of 06:22, 14 July 2010 (view source)← Older edit Revision as of 06:24, 14 July 2010 (view source)Newer edit → Line 118: Line 118: |} |} - Substituting the above four integrated total fluxes into eq. (4.263), we have + Substituting the above four integrated total fluxes into eq. (5), we have {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 129: Line 129: \end{align}[/itex] \end{align}[/itex] - |{{EquationRef|(5)}} + |{{EquationRef|(6)}} |} |} where where Line 145: Line 145: $a_{P}\varphi _{P}=a_{E}\varphi _{E}+a_{W}\varphi _{W}+a_{N}\varphi _{N}+a_{S}\varphi _{S}+b$ $a_{P}\varphi _{P}=a_{E}\varphi _{E}+a_{W}\varphi _{W}+a_{N}\varphi _{N}+a_{S}\varphi _{S}+b$ - |{{EquationRef|(6)}} + |{{EquationRef|(7)}} |} |} where where Line 154: Line 154: $a_{E}=D_{e}A(\text{Pe}_{\Delta e})=D_{e}\left\{ A(\left| \text{Pe}_{\Delta e} \right|)+\left[\!\left[ -\text{Pe}_{\Delta e},0 \right]\!\right] \right\}$ $a_{E}=D_{e}A(\text{Pe}_{\Delta e})=D_{e}\left\{ A(\left| \text{Pe}_{\Delta e} \right|)+\left[\!\left[ -\text{Pe}_{\Delta e},0 \right]\!\right] \right\}$ - |{{EquationRef|(7)}} + |{{EquationRef|(8)}} |} |} Line 162: Line 162: $a_{W}=D_{w}B(\text{Pe}_{\Delta w})=D_{w}\left\{ A(\left| \text{Pe}_{\Delta w} \right|)+\left[\!\left[ \text{Pe}_{\Delta w},0 \right]\!\right] \right\}$ $a_{W}=D_{w}B(\text{Pe}_{\Delta w})=D_{w}\left\{ A(\left| \text{Pe}_{\Delta w} \right|)+\left[\!\left[ \text{Pe}_{\Delta w},0 \right]\!\right] \right\}$ - |{{EquationRef|(8)}} + |{{EquationRef|(9)}} |} |} Line 170: Line 170: $a_{N}=D_{n}A(\text{Pe}_{\Delta n})=D_{n}\left\{ A(\left| \text{Pe}_{\Delta n} \right|)+\left[\!\left[ -\text{Pe}_{\Delta n},0 \right]\!\right] \right\}$ $a_{N}=D_{n}A(\text{Pe}_{\Delta n})=D_{n}\left\{ A(\left| \text{Pe}_{\Delta n} \right|)+\left[\!\left[ -\text{Pe}_{\Delta n},0 \right]\!\right] \right\}$ - |{{EquationRef|(9)}} + |{{EquationRef|(10)}} |} |} Line 178: Line 178: $a_{S}=D_{s}B(\text{Pe}_{\Delta s})=D_{s}\left\{ A(\left| \text{Pe}_{\Delta s} \right|)+\left[\!\left[ \text{Pe}_{\Delta s},0 \right]\!\right] \right\}$ $a_{S}=D_{s}B(\text{Pe}_{\Delta s})=D_{s}\left\{ A(\left| \text{Pe}_{\Delta s} \right|)+\left[\!\left[ \text{Pe}_{\Delta s},0 \right]\!\right] \right\}$ - |{{EquationRef|(10)}} + |{{EquationRef|(11)}} |} |} Line 187: Line 187: - |{{EquationRef|(11)}} + |{{EquationRef|(12)}} |} |} Line 198: Line 198: \end{align}[/itex] \end{align}[/itex] - | {{EquationRef|(12)}} + | {{EquationRef|(13)}} |} |} Line 209: Line 209: - |{{EquationRef|(13)}} + |{{EquationRef|(14)}} |} |} If the continuity equation is satisfied, eq. (4.271) can be simplified as (see Problem 4.4) If the continuity equation is satisfied, eq. (4.271) can be simplified as (see Problem 4.4) Line 218: Line 218: $a_{P}=a_{E}+a_{W}+a_{N}+a_{S}+a_{P}^{0}-S_{P}\Delta x\Delta y$ $a_{P}=a_{E}+a_{W}+a_{N}+a_{S}+a_{P}^{0}-S_{P}\Delta x\Delta y$ - |{{EquationRef|(14)}} + |{{EquationRef|(15)}} |} |} Similar to the case of one-dimensional convection-diffusion, different discretization schemes for the discretized equations (4.265) – (4.272) can be obtained by using different expressions for A(|PeΔ|) from Table 4.3. Similar to the case of one-dimensional convection-diffusion, different discretization schemes for the discretized equations (4.265) – (4.272) can be obtained by using different expressions for A(|PeΔ|) from Table 4.3. Line 230: Line 230: $a_{P}\varphi _{P}=a_{E}\varphi _{E}+a_{W}\varphi _{W}+a_{N}\varphi _{N}+a_{S}\varphi _{S}+a_{T}\varphi _{T}+a_{B}\varphi _{B}+b$ $a_{P}\varphi _{P}=a_{E}\varphi _{E}+a_{W}\varphi _{W}+a_{N}\varphi _{N}+a_{S}\varphi _{S}+a_{T}\varphi _{T}+a_{B}\varphi _{B}+b$ - |{{EquationRef|(15)}} + |{{EquationRef|(16)}} |} |} where where Line 239: Line 239: $a_{E}=D_{e}\left\{ A(\left| \text{Pe}_{\Delta e} \right|)+\left[\!\left[ -\text{Pe}_{\Delta e},0 \right]\!\right] \right\}$ $a_{E}=D_{e}\left\{ A(\left| \text{Pe}_{\Delta e} \right|)+\left[\!\left[ -\text{Pe}_{\Delta e},0 \right]\!\right] \right\}$ - |{{EquationRef|(16)}} + |{{EquationRef|(17)}} |} |} Line 247: Line 247: $a_{W}=D_{w}\left\{ A(\left| \text{Pe}_{\Delta w} \right|)+\left[\!\left[ \text{Pe}_{\Delta w},0 \right]\!\right] \right\}$ $a_{W}=D_{w}\left\{ A(\left| \text{Pe}_{\Delta w} \right|)+\left[\!\left[ \text{Pe}_{\Delta w},0 \right]\!\right] \right\}$ - |{{EquationRef|(17)}} + |{{EquationRef|(18)}} |} |} Line 255: Line 255: $a_{N}=D_{n}\left\{ A(\left| \text{Pe}_{\Delta n} \right|)+\left[\!\left[ -\text{Pe}_{\Delta n},0 \right]\!\right] \right\}$ $a_{N}=D_{n}\left\{ A(\left| \text{Pe}_{\Delta n} \right|)+\left[\!\left[ -\text{Pe}_{\Delta n},0 \right]\!\right] \right\}$ - |{{EquationRef|(18)}} + |{{EquationRef|(19)}} |} |} Line 263: Line 263: $a_{S}=D_{s}\left\{ A(\left| \text{Pe}_{\Delta s} \right|)+\left[\!\left[ \text{Pe}_{\Delta s},0 \right]\!\right] \right\}$ $a_{S}=D_{s}\left\{ A(\left| \text{Pe}_{\Delta s} \right|)+\left[\!\left[ \text{Pe}_{\Delta s},0 \right]\!\right] \right\}$ - |{{EquationRef|(19)}} + |{{EquationRef|(20)}} |} |} Line 271: Line 271: $a_{T}=D_{t}\left\{ A(\left| \text{Pe}_{\Delta t} \right|)+\left[\!\left[ -\text{Pe}_{\Delta t},0 \right]\!\right] \right\}$ $a_{T}=D_{t}\left\{ A(\left| \text{Pe}_{\Delta t} \right|)+\left[\!\left[ -\text{Pe}_{\Delta t},0 \right]\!\right] \right\}$ - |{{EquationRef|(20)}} + |{{EquationRef|(21)}} |} |} Line 279: Line 279: $a_{B}=D_{b}\left\{ A(\left| \text{Pe}_{\Delta b} \right|)+\left[\!\left[ \text{Pe}_{\Delta b},0 \right]\!\right] \right\}$ $a_{B}=D_{b}\left\{ A(\left| \text{Pe}_{\Delta b} \right|)+\left[\!\left[ \text{Pe}_{\Delta b},0 \right]\!\right] \right\}$ - |{{EquationRef|(21)}} + |{{EquationRef|(22)}} |} |} Line 288: Line 288: - |{{EquationRef|(22)}} + |{{EquationRef|(23)}} |} |} Line 296: Line 296: $a_{P}=a_{E}+a_{W}+a_{N}+a_{S}+a_{T}+a_{B}+a_{P}^{0}-S_{P}\Delta x\Delta y\Delta z$ $a_{P}=a_{E}+a_{W}+a_{N}+a_{S}+a_{T}+a_{B}+a_{P}^{0}-S_{P}\Delta x\Delta y\Delta z$ - |{{EquationRef|(23)}} + |{{EquationRef|(24)}} |} |} Line 305: Line 305: - |{{EquationRef|(24)}} + |{{EquationRef|(25)}} |} |} The expressions for conductance at the faces of the control volume are The expressions for conductance at the faces of the control volume are Line 316: Line 316: \end{align}[/itex] \end{align}[/itex] - | {{EquationRef|(25)}} + | {{EquationRef|(26)}} |} |} Line 329: Line 329: \end{align}[/itex] \end{align}[/itex] - | {{EquationRef|(26)}} + | {{EquationRef|(27)}} |} |} The Different discretization schemes for the above three-dimensional problem can be obtained by using different expressions for A(|PeΔ|) from Table 4.3. The Different discretization schemes for the above three-dimensional problem can be obtained by using different expressions for A(|PeΔ|) from Table 4.3. In addition to the six first order discretization schemes described above, some researchers have used higher order schemes such as second order upwind (Leonard et al., 1981) and QUICK (Quadratic Upwind Interpolation of Convective Kinetics; Leonard, 1979) schemes to overcome the false diffusion problem, which is referred to as error caused by using the discretization scheme with accuracy less than the second order (Patankar, 1980). The error due to false diffusion could potentially be severe for (1) transient problems, (2) multidimensional steady-state problems, or (3) problems with non-constant source terms (Tao, 2001). While the accuracies of these higher order schemes are better than the first order schemes, their computational time is much greater than that of the first order schemes. In addition to the six first order discretization schemes described above, some researchers have used higher order schemes such as second order upwind (Leonard et al., 1981) and QUICK (Quadratic Upwind Interpolation of Convective Kinetics; Leonard, 1979) schemes to overcome the false diffusion problem, which is referred to as error caused by using the discretization scheme with accuracy less than the second order (Patankar, 1980). The error due to false diffusion could potentially be severe for (1) transient problems, (2) multidimensional steady-state problems, or (3) problems with non-constant source terms (Tao, 2001). While the accuracies of these higher order schemes are better than the first order schemes, their computational time is much greater than that of the first order schemes.

## Two-dimensional problem

The heat transfer problems discussed in the preceding subsection are steady-state convection-diffusion problems with the general variable $\varphi$ varying in one dimension only. We now turn our attention to the unsteady state two-dimensional convection-diffusion problem which includes a source term S. The problem is described by $\frac{\partial (\rho \varphi )}{\partial t}+\frac{\partial (\rho u\varphi )}{\partial x}+\frac{\partial (\rho v\varphi )}{\partial y}=\frac{\partial }{\partial x}\left( \Gamma \frac{\partial \varphi }{\partial x} \right)+\frac{\partial }{\partial y}\left( \Gamma \frac{\partial \varphi }{\partial y} \right)+S$ (1)

which can be rewritten as $\frac{\partial (\rho \varphi )}{\partial t}+\frac{\partial J_{x}}{\partial x}+\frac{\partial J_{y}}{\partial y}=S$ (2)

where $J_{x}=\rho u\varphi -\Gamma \frac{\partial \varphi }{\partial x}$ (3) $J_{y}=\rho v\varphi -\Gamma \frac{\partial \varphi }{\partial y}$ (4)

Integrating eq. (2) with respect to t in the interval of (t, t+Δt) and over the control volume P, we have \begin{align} & \int_{s}^{n}{\int_{e}^{w}{\int_{t}^{t+\Delta t}{\frac{\partial }{\partial t}\left( \rho \varphi \right)dt}dxdy}}+\int_{t}^{t+\Delta t}{\int_{s}^{n}{\int_{w}^{e}{\frac{\partial J_{x}}{\partial x}dxdydt}}}+\int_{t}^{t+\Delta t}{\int_{e}^{w}{\int_{s}^{n}{\frac{\partial J_{y}}{\partial y}dydxdt}}} \\ & =\int_{t}^{t+\Delta t}{\int_{s}^{n}{\int_{w}^{e}{(S_{C}+S_{P}\varphi )dxdydt}}} \\ \end{align}

where the source term is treated as a linear function of $\varphi$. Assuming the total fluxes are uniform on all faces of the control volume and employing fully-implicit scheme, the above equation becomes $(\rho _{P}\varphi _{P}-\rho _{P}^{0}\varphi _{P}^{0})\Delta x\Delta y+(J_{x}^{e}-J_{x}^{w})\Delta y\Delta t+(J_{y}^{n}-J_{y}^{s})\Delta x\Delta t=(S_{C}+S_{P}\varphi _{P})\Delta x\Delta y\Delta t$

where the superscript 0 represents the values at the previous time step. Introducing the integrated total fluxes $J_{e}=J_{x}^{e}\Delta y$, $J_{w}=J_{x}^{w}\Delta y$, $J_{n}=J_{y}^{n}\Delta x$ and $J_{s}=J_{y}^{s}\Delta x$, and dividing the above equation by Δt yields $\frac{(\rho _{P}\varphi _{P}-\rho _{P}^{0}\varphi _{P}^{0})}{\Delta t}\Delta x\Delta y+(J_{e}-J_{w})+(J_{n}-J_{s})=(S_{C}+S_{P}\varphi _{P})\Delta x\Delta y$ (5)

Defining the following integrated total flux $J_{e}^{*}=\frac{J_{e}}{D_{e}},\text{ }J_{w}^{*}=\frac{J_{w}}{D_{w}},\text{ }J_{n}^{*}=\frac{J_{n}}{D_{n}},\text{ }J_{s}^{*}=\frac{J_{s}}{D_{s}}$

where $D_{e}=\frac{\Gamma _{e}}{(\delta x)_{e}}\Delta y,\text{ }D_{w}=\frac{\Gamma _{w}}{(\delta x)_{w}}\Delta y,\text{ }D_{n}=\frac{\Gamma _{n}}{(\delta y)_{n}}\Delta x,\text{ }D_{s}=\frac{\Gamma _{s}}{(\delta y)_{s}}\Delta x$

the integrated fluxes at the east and west faces of the control volume can be evaluated: $J_{e}=D_{e}J_{e}^{*}=D_{e}[B(\text{Pe}_{\Delta e})\varphi _{P}-A(\text{Pe}_{\Delta e})\varphi _{E}]$ $J_{w}=D_{w}J_{w}^{*}=D_{w}[B(\text{Pe}_{\Delta w})\varphi _{W}-A(\text{Pe}_{\Delta w})\varphi _{P}]$

Substituting B(PeΔ) − A(PeΔ) = PeΔ into the two equations above yields $J_{e}=D_{e}J_{e}^{*}=D_{e}[A(\text{Pe}_{\Delta e})\varphi _{P}+\text{Pe}_{\Delta e}\varphi _{P}-A(\text{Pe}_{\Delta e})\varphi _{E}]$ $J_{w}=D_{w}J_{w}^{*}=D_{w}[B(\text{Pe}_{\Delta w})\varphi _{W}-B(\text{Pe}_{\Delta w})\varphi _{P}+\text{Pe}_{\Delta w}\varphi _{P}]$

Similarly, the integrated total flux at the north and south faces of the control volume can be expressed as $J_{n}=D_{n}J_{n}^{*}=D_{n}[A(\text{Pe}_{\Delta n})\varphi _{P}+\text{Pe}_{\Delta n}\varphi _{P}-A(\text{Pe}_{\Delta n})\varphi _{N}]$ $J_{s}=D_{s}J_{s}^{*}=D_{s}[B(\text{Pe}_{\Delta s})\varphi _{S}-B(\text{Pe}_{\Delta s})\varphi _{P}+\text{Pe}_{\Delta s}\varphi _{P}]$

Substituting the above four integrated total fluxes into eq. (5), we have \begin{align} & \left\{ \rho _{P}\Delta x\Delta y/\Delta t-S_{P}\Delta x\Delta y+D_{e}A(\text{Pe}_{\Delta e})+D_{w}B(\text{Pe}_{\Delta w}) \right. \\ & \left. +D_{n}A(\text{Pe}_{\Delta n})+D_{s}B(\text{Pe}_{\Delta s})+(F_{e}-F_{w})+(F_{n}-F_{s}) \right\}\varphi _{P} \\ & =D_{e}A(\text{Pe}_{\Delta e})\varphi _{E}+D_{w}B(\text{Pe}_{\Delta w})\varphi _{W}+D_{n}A(\text{Pe}_{\Delta n})\varphi _{N} \\ & +D_{s}B(\text{Pe}_{\Delta s})\varphi _{S}+S_{C}\Delta x\Delta y+(\rho _{P}^{0}\Delta x\Delta y/\Delta t)\varphi _{P}^{0} \\ \end{align} (6)

where $\begin{matrix}{}\\\end{matrix}F_{e}=D_{e}\text{Pe}_{\Delta e}=(\rho u)_{e}\Delta y,\text{ }F_{w}=D_{w}\text{Pe}_{\Delta w}=(\rho u)_{w}\Delta y$ $\begin{matrix}{}\\\end{matrix}F_{n}=D_{n}\text{Pe}_{\Delta n}=(\rho v)_{n}\Delta x,\text{ }F_{s}=D_{s}\text{Pe}_{\Delta s}=(\rho v)_{s}\Delta x$

are the mass flow rates at the four faces of the control volume. Equation (4.264) can be rearranged to obtain the final form of the following discretized equation: $a_{P}\varphi _{P}=a_{E}\varphi _{E}+a_{W}\varphi _{W}+a_{N}\varphi _{N}+a_{S}\varphi _{S}+b$ (7)

where $a_{E}=D_{e}A(\text{Pe}_{\Delta e})=D_{e}\left\{ A(\left| \text{Pe}_{\Delta e} \right|)+\left[\!\left[ -\text{Pe}_{\Delta e},0 \right]\!\right] \right\}$ (8) $a_{W}=D_{w}B(\text{Pe}_{\Delta w})=D_{w}\left\{ A(\left| \text{Pe}_{\Delta w} \right|)+\left[\!\left[ \text{Pe}_{\Delta w},0 \right]\!\right] \right\}$ (9) $a_{N}=D_{n}A(\text{Pe}_{\Delta n})=D_{n}\left\{ A(\left| \text{Pe}_{\Delta n} \right|)+\left[\!\left[ -\text{Pe}_{\Delta n},0 \right]\!\right] \right\}$ (10) $a_{S}=D_{s}B(\text{Pe}_{\Delta s})=D_{s}\left\{ A(\left| \text{Pe}_{\Delta s} \right|)+\left[\!\left[ \text{Pe}_{\Delta s},0 \right]\!\right] \right\}$ (11) $b=a_{P}^{0}\varphi _{P}^{0}+S_{C}\Delta x\Delta y$ (12) \begin{align} & a_{P}=a_{E}+a_{W}+a_{N}+a_{S}+(\rho _{P}/\rho _{P}^{0})a_{P}^{0} \\ & \text{ }-S_{P}\Delta x\Delta y+(F_{e}-F_{w})+(F_{n}-F_{s}) \\ \end{align} (13) $a_{P}^{0}=\frac{\rho _{P}^{0}\Delta x\Delta y}{\Delta t}$ (14)

If the continuity equation is satisfied, eq. (4.271) can be simplified as (see Problem 4.4) $a_{P}=a_{E}+a_{W}+a_{N}+a_{S}+a_{P}^{0}-S_{P}\Delta x\Delta y$ (15)

Similar to the case of one-dimensional convection-diffusion, different discretization schemes for the discretized equations (4.265) – (4.272) can be obtained by using different expressions for A(|PeΔ|) from Table 4.3.

## Three-dimensional problem

The discretized equation for a transient three-dimensional convection-diffusion problem can be obtained by integrating the conservation equation with respect to t in the interval of (t, t+Δt) and over the three-dimensional control volume P (formed by considering two additional neighbors at top, T, and bottom, B). The final form of the governing equation is (Patankar, 1980) $a_{P}\varphi _{P}=a_{E}\varphi _{E}+a_{W}\varphi _{W}+a_{N}\varphi _{N}+a_{S}\varphi _{S}+a_{T}\varphi _{T}+a_{B}\varphi _{B}+b$ (16)

where $a_{E}=D_{e}\left\{ A(\left| \text{Pe}_{\Delta e} \right|)+\left[\!\left[ -\text{Pe}_{\Delta e},0 \right]\!\right] \right\}$ (17) $a_{W}=D_{w}\left\{ A(\left| \text{Pe}_{\Delta w} \right|)+\left[\!\left[ \text{Pe}_{\Delta w},0 \right]\!\right] \right\}$ (18) $a_{N}=D_{n}\left\{ A(\left| \text{Pe}_{\Delta n} \right|)+\left[\!\left[ -\text{Pe}_{\Delta n},0 \right]\!\right] \right\}$ (19) $a_{S}=D_{s}\left\{ A(\left| \text{Pe}_{\Delta s} \right|)+\left[\!\left[ \text{Pe}_{\Delta s},0 \right]\!\right] \right\}$ (20) $a_{T}=D_{t}\left\{ A(\left| \text{Pe}_{\Delta t} \right|)+\left[\!\left[ -\text{Pe}_{\Delta t},0 \right]\!\right] \right\}$ (21) $a_{B}=D_{b}\left\{ A(\left| \text{Pe}_{\Delta b} \right|)+\left[\!\left[ \text{Pe}_{\Delta b},0 \right]\!\right] \right\}$ (22) $b=a_{P}^{0}\varphi _{P}^{0}+S_{C}\Delta x\Delta y\Delta z$ (23) $a_{P}=a_{E}+a_{W}+a_{N}+a_{S}+a_{T}+a_{B}+a_{P}^{0}-S_{P}\Delta x\Delta y\Delta z$ (24) $a_{P}^{0}=\frac{\rho \Delta x\Delta y\Delta z}{\Delta t}$ (25)

The expressions for conductance at the faces of the control volume are \begin{align} & D_{e}=\frac{\Gamma _{e}\Delta y\Delta z}{(\delta x)_{e}},\text{ }D_{w}=\frac{\Gamma _{w}\Delta y\Delta z}{(\delta x)_{w}},\text{ }D_{n}=\frac{\Gamma _{n}\Delta x\Delta z}{(\delta y)_{n}} \\ & D_{s}=\frac{\Gamma _{s}\Delta x\Delta z}{(\delta y)_{s}},\text{ }D_{t}=\frac{\Gamma _{t}\Delta x\Delta y}{(\delta z)_{t}},\text{ }D_{b}=\frac{\Gamma _{b}\Delta x\Delta y}{(\delta z)_{b}} \\ \end{align} (26)

and the flow rates are: \begin{align} & F_{e}=D_{e}\text{Pe}_{\Delta e}=(\rho u)_{e}\Delta y\Delta z,\text{ }F_{w}=D_{w}\text{Pe}_{\Delta w}=(\rho u)_{w}\Delta y\Delta z \\ & F_{n}=D_{n}\text{Pe}_{\Delta n}=(\rho v)_{n}\Delta x\Delta z,\text{ }F_{s}=D_{s}\text{Pe}_{\Delta s}=(\rho v)_{s}\Delta x\Delta z \\ & F_{t}=D_{t}\text{Pe}_{\Delta t}=(\rho w)_{t}\Delta x\Delta y,\text{ }F_{b}=D_{b}\text{Pe}_{\Delta b}=(\rho w)_{b}\Delta x\Delta y \\ \end{align} (27)

The Different discretization schemes for the above three-dimensional problem can be obtained by using different expressions for A(|PeΔ|) from Table 4.3. In addition to the six first order discretization schemes described above, some researchers have used higher order schemes such as second order upwind (Leonard et al., 1981) and QUICK (Quadratic Upwind Interpolation of Convective Kinetics; Leonard, 1979) schemes to overcome the false diffusion problem, which is referred to as error caused by using the discretization scheme with accuracy less than the second order (Patankar, 1980). The error due to false diffusion could potentially be severe for (1) transient problems, (2) multidimensional steady-state problems, or (3) problems with non-constant source terms (Tao, 2001). While the accuracies of these higher order schemes are better than the first order schemes, their computational time is much greater than that of the first order schemes.