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Governing equations; forward, backward, central differencing

Short answer

Fluid movement is governed by the Saint-Venant equations: continuity ∂A/∂t+∂Q/∂x=q and momentum with local/convective acceleration, pressure, gravity and friction. Finite difference approximations: forward (f_{i+1}-f_i)/Δx, backward (f_i-f_{i-1})/Δx (both first-order accurate); central (f_{i+1}-f_{i-1})/(2Δx) (second-order).

Write down the governing equations used for analyzing the movement of fluid. Discuss forward, backward and central differencing with expressions.

Governing equations of fluid movement

The unsteady flow of water in an open channel is governed by the Saint-Venant (1D shallow water) equations.

Continuity (conservation of mass):

∂A∂t+∂Q∂x=q\frac{\partial A}{\partial t}+\frac{\partial Q}{\partial x}=q

Momentum (conservation of momentum):

∂Q∂t⏟local acc.+∂∂x ⁣(Q2A)⏟convective acc.+gA∂y∂x⏟pressure force−gAS0⏟gravity+gASf⏟friction=0\underbrace{\frac{\partial Q}{\partial t}}_{\text{local acc.}}+\underbrace{\frac{\partial}{\partial x}\!\left(\frac{Q^2}{A}\right)}_{\text{convective acc.}}+\underbrace{gA\frac{\partial y}{\partial x}}_{\text{pressure force}}-\underbrace{gAS_0}_{\text{gravity}}+\underbrace{gAS_f}_{\text{friction}}=0

where AA = flow area, QQ = discharge, yy = flow depth, qq = lateral inflow per unit length, S0S_0 = bed slope, Sf=n2Q∣Q∣A2R4/3S_f=\dfrac{n^2Q|Q|}{A^2R^{4/3}} = friction slope (Manning), gg = gravity. These are the Saint-Venant equations (1D unsteady open-channel flow).

In terms of velocity VV and depth yy for a wide rectangular channel:

∂y∂t+V∂y∂x+y∂V∂x=0,∂V∂t+V∂V∂x+g∂y∂x=g(S0−Sf)\frac{\partial y}{\partial t}+V\frac{\partial y}{\partial x}+y\frac{\partial V}{\partial x}=0,\qquad \frac{\partial V}{\partial t}+V\frac{\partial V}{\partial x}+g\frac{\partial y}{\partial x}=g(S_0-S_f)

Forward, backward and central differencing

The grid has points xi=iΔxx_i=i\Delta x. The derivative at point ii is approximated using neighbouring values.

Expand ff about xix_i by Taylor's series:

fi+1=fi+Δxfi′+Δx22fi′′+Δx36fi(3)+…,fi−1=fi−Δxfi′+Δx22fi′′−Δx36fi(3)+…f_{i+1}=f_i+\Delta x f'_i+\frac{\Delta x^2}{2}f''_i+\frac{\Delta x^3}{6}f_i^{(3)}+\dots,\qquad f_{i-1}=f_i-\Delta x f'_i+\frac{\Delta x^2}{2}f''_i-\frac{\Delta x^3}{6}f_i^{(3)}+\dots
SchemeExpressionError
Forward$\left.\dfrac{\partial f}{\partial x}\righti\approx\dfrac{f{i+1}-f_i}{\Delta x}$
Backward$\left.\dfrac{\partial f}{\partial x}\righti\approx\dfrac{f_i-f{i-1}}{\Delta x}$
Central$\left.\dfrac{\partial f}{\partial x}\righti\approx\dfrac{f{i+1}-f_{i-1}}{2\Delta x}$
Second derivative$\left.\dfrac{\partial^2 f}{\partial x^2}\righti\approx\dfrac{f{i+1}-2f_i+f_{i-1}}{\Delta x^2}$

The same formulas are used for the time derivative with Δt\Delta t (e.g. forward in time: ∂Q/∂t≈(Qin+1−Qin)/Δt\partial Q/\partial t\approx (Q_i^{n+1}-Q_i^n)/\Delta t).

 f
 |                     . f(i+1)
 |                 . '
 |            f(i)
 |         .'
 |   f(i-1)
 +----|---------|---------|----> x
     i-1        i        i+1

 forward : chord i  -> i+1
 backward: chord i-1-> i
 central : chord i-1-> i+1 (closest to tangent at i)
  • Forward uses the point ahead; backward uses the point behind (suited to flow from upstream, "upwind"); central uses both neighbours and is more accurate, but is less stable for explicit time marching.

✓ Answer checked by Claude Opus 5.5 (high effort)

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