doc/algorithm/finitevol-meth.rst

4f2617d475 Jeff*0001 The finite volume method: finite volumes versus finite difference
                0002 -----------------------------------------------------------------
                0003
                0004 The finite volume method is used to discretize the equations in space.
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0005 The expression “finite volume” actually has two meanings; one is the
                0006 method of embedded or intersecting boundaries (shaved or lopped cells in
                0007 our terminology) and the other is non-linear interpolation methods that
                0008 can deal with non-smooth solutions such as shocks (i.e. flux limiters
                0009 for advection). Both make use of the integral form of the conservation
                0010 laws to which the *weak solution* is a solution on each finite volume of
                0011 (sub-domain). The weak solution can be constructed out of piece-wise
                0012 constant elements or be differentiable. The differentiable equations can
                0013 not be satisfied by piece-wise constant functions.
                0014
                0015 As an example, the 1-D constant coefficient advection-diffusion
                0016 equation:
                0017
                0018 .. math:: \partial_t \theta + \partial_x ( u \theta - \kappa \partial_x \theta ) = 0
                0019
                0020 can be discretized by integrating over finite sub-domains, i.e. the
                0021 lengths :math:`\Delta x_i`:
                0022
                0023 .. math:: \Delta x \partial_t \theta + \delta_i ( F ) = 0
                0024
                0025 is exact if :math:`\theta(x)` is piece-wise constant over the interval
                0026 :math:`\Delta x_i` or more generally if :math:`\theta_i` is defined as
                0027 the average over the interval :math:`\Delta x_i`.
                0028
                0029 The flux, :math:`F_{i-1/2}`, must be approximated:
                0030
                0031 .. math:: F = u \overline{\theta} - \frac{\kappa}{\Delta x_c} \partial_i \theta
                0032
                0033 and this is where truncation errors can enter the solution. The method
                0034 for obtaining :math:`\overline{\theta}` is unspecified and a wide range
                0035 of possibilities exist including centered and upwind interpolation,
                0036 polynomial fits based on the the volume average definitions of
                0037 quantities and non-linear interpolation such as flux-limiters.
                0038
                0039 Choosing simple centered second-order interpolation and differencing
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0040 recovers the same ODE’s resulting from finite differencing for the
                0041 interior of a fluid. Differences arise at boundaries where a boundary is
                0042 not positioned on a regular or smoothly varying grid. This method is
                0043 used to represent the topography using lopped cell, see Adcroft et al. (1997)
                0044 :cite:`adcroft:97`. Subtle difference also appear in more
                0045 than one dimension away from boundaries. This happens because each
                0046 direction is discretized independently in the finite difference method
                0047 while the integrating over finite volume implicitly treats all
                0048 directions simultaneously.