1.4.0
catchment modelling framework
Finite Volume Method

Introduction

This page describes the theory of the finite volume method and the application the cmf architecture. Mass conservation

General mass conservation problem

Any transport problem is largely governed by mass conservation. The closure of the mass balance in a hydrologic model should be the starting point of all model applications, even though the closure cannot be routinely verified by field experiments. Mass conservative problems in space, including water and solute transport in the landscape, can be expressed as a generalized hyperbolic problem:

\[ \frac{{\delta {\bf{u}}} }{{\delta t}} + \nabla \cdot f\left( {{\bf{u}},t} \right) = 0 \hspace{20mm}(1) \]

with the state continuum \(\bf{u}\), and the fluxes \(f({\bf{u}},t)\).

Finite volume discretization

Using the finite volume method for discretizing u space, the partial differential equation turns into a system of ordinary differential equations (ODE's), where the fluxes in and out of the finite volume \(i\) are integrated over the shell of the volume \(S\) and normalized by the size of the finite volume \(V_i\).

\[ \frac{{du_i }}{{dt}} = - \frac{{\oint_{S_i } {f({\bf{u}},t)dS} }}{{v_i }} \hspace{20mm}(2) \]

If the shell is constructed by a finite number of planes, the line integral over the shell is transformed to the sum of fluxes connecting adjacent volumes. When the FVM-discretization is applied to a water and solute problem, a wide range of model approaches can be implemented. The model approaches differ by the method used to discretize the problem domain, the flux functions (e.g. Richard's equation or linear storage function) and by the complexity of connections between the finite volumes.

The CMF approach covers the flexibility of the description of volumes and fluxes by providing clearly defined and relatively independent interfaces to implement flux functions and discretization methods. A range from highly detailed one, two or three dimensional plot scale water and solute transport models to regional applications using simplified flux descriptions and discretization / connection concepts is possible.

The abstraction hierarchy

One of the keys for an open extensible framework is the abstraction of common behavior of different objects in a model. In CMF, different layers of abstraction exist to facilitate the access to processes and data by a single interface. By definition, the abstract nature of the interface results in a situation where the type of process and how it is described is of no consequence to the simulation framework. It is of course fundamentally important to the user. Two primary abstraction hierarchies have been implemented as part of the core CMF structure: storages and fluxes.

Storage abstraction

Storages, at the most abstract layer, are defined as state variables and expose their state and a function to calculate their derivatives at a given time. The state of an individual water storage is the volume of water and the change rate is the sum of the fluxes in and out of the volume. The water budget of a general water storage, derived from (2) is given by (3).

\[ \frac{{dV_i }}{{dt}} = \sum\limits_{j = 1}^{N_i } {\left( { - q_{i,j} \left( {V_i ,V_j ,t} \right)A_{i,j} } \right)} \hspace{20mm}(3) \]

\(V\) is the volume of stored water in the control volume, \(i\) the current control volume, \(N\) the number of connected storages to \(i\), \(q_{i,j}\) the flux from \(i\) to \(j\), and \(A_{i,j}\) is the cross sectional area of the flux.

Solute transport

The state of a solute storage is the amount of tracer particles in the finite volume. The change rate of the state is the sum of advective tracer fluxes and the sum of any existent source or sink flux.

\[ \frac{{dX_i }}{{dt}} = \sum\limits_{j = 1}^{N_i } {\left( { - q_{i,j} \left( {V_i ,V_j ,t} \right)[X]_s } \right) + q_{X,in} \left( t \right) - r^ - X} \hspace{20mm}(4) \]

\(X_i\) is the amount (mols or mass) of a tracer in control volume \(i\), \([X]\) is the concentration of the tracer in amount per volume water and \(q_{X,in}(t)\) is a source or sink flux in amount per day (eg. g/day) and \(r^-\) is a decay rate in 1/day. \([X]_s\) is the concentration of the source of the flux, determined by the sign of \(q_{i,j}\).

A water store maintains a reference to each tracer store for each simulated tracer. The figure shows the storage concept hierarchy in CMF. If needed, state variables for momentum or energy conservation models could be introduced.

To facilitate the calculation of fluxes, specializations of water storages are implemented. Open water storages provide a calculation of their fill height as a function of state, and soil water storages have additional values to characterize the position in the soil column and the water retention curve.

The retention curve is an object with methods calculating conductivity and matrix potential from a given water saturation of a soil water storage.

Using the storages, a model of the landscape is constructed, as described in Building a landscape model .

Flux abstraction

Fluxes of water in CMF are modeled as a flow network, where the fluxes occur at the edges of the network. The nodes of the network are the water storages and boundary conditions, while the edges are represented by flux connection objects (s.Figure 2). The fluxes are calculated using specializations of the flux connections, representing the mathematical models of the hydrologic behaviour of storage interaction.

Each flux node maintains a method to calculate its water balance, as the sum of the in- and outgoing fluxes. For water storages, the water balance equals the derivative of the stored water volume, while the water balance of other flux node types can be queried to calculate the boundary fluxes between distinct subsystems and to calculate the area water budget.

Each specialization of a flux connection is the implementation of a flux generating process. In the current version of CMF, mostly well known and tested processes are implemented. These include, among others, the Richards equation adopted from the BROOK 90 model, a tipping bucket model adopted from SWAT, a linear storage, the Manning equation and the Penman-Monteith and Hargreave equations to define evapotransporation.

Since the implementation of a flux generating process is relatively simple, and because a set of well-established numerical procedures for the solution of systems of differential are integrated into the core model components, only few lines of code are required to implement additional processes. The Richards equation, for example, needs only 20 lines of code, including the handling of special cases and all overhead associated with the CMF infrastructure. Thus the extension of CMF by new process descriptions, like handling macro pores, is relatively simple assuming one has a clear mathematical description of the process of interest.

A simplified class model of cmf is shown here: CmfSoftwareObjects, while a complete list of available classes is here

Mathematical solving system

The state variables provide methods for calculating their derivatives, thus any collection of state variables forms a system of ordinary differential equations, to be integrated by any well known integration scheme for initial value problems (IVP).

All solvers, except the CVODE solver are implemented using OpenMP for parallel computation of the derivatives, while the original CVODE code is integrated into CMF as a static library. The CVODE library is also parallelized using OpenMP. The selection of the most appropriate solver for the ODEs making up any one model is not trivial. While explicit methods, like the explicit Euler scheme and the Runge-Kutta-Fehlberg (RKF) method are computationally inexpensive per time step, many possible setups of CMF are too unstable, or stiff, to be solved by an explicit method. The stiffer the system, the more complex integration method needs to be used. Stiffness of systems is indicated by the appearance of processes with highly different time scales. This is especially the case if the Richards equation is to be solved and surface runoff is allowed to reinfiltrate the soil. For such systems only the CVODE solver with an appropriate preconditioner for the Newton-Krylov iteration is able to solve the system. While CMF cannot provide significant guidance in terms of the most appropriate solution procedure, the fact that a variety of choices is available has a large degree of utility for modellers with an interest in the process of discretizing continuous systems of equations to develop solutions, and in particular how those choices may be reflected in modelled results. Again, we make the argument that the most appropriate model for any unique situation may best be arrived through some degree of experimentation. The solution procedure is part of this process, and CMF is designed to explicitly recognize this fact.