Appendix D: Details of FVM implementations of diffusion term This section details the methods used to approximate the diffusion coefficient at the control volume boundaries when the diffusion coefficient is piecewise constant and either over the entire domain or known only at the nodes. Average of node value This standard method uses simple linear interpolation to estimate the diffusion coefficient at the control volume boundaries using the values of at the adjacent nodes D [9-10]. It is implemented here in situations where is only known at the nodes but is D also used where is piecewise constant. In the piecewise-constant case, it makes no use D of the fact that is known between the nodes. The general equation for the diffusion D(x) coefficient at the boundary between nodes and is n n+1 the superscript refers to of the node values”) which, for uniform node An “average spacing, simplifies to Average of average values The second non-standard approach denoted using a superscript and only used where AA, is piecewise-constant, takes account of the diffusion coefficient at all points by first D calculating the average values of over each volume using D The diffusion coefficient at a control volume boundary is then calculated as Average between the nodes The non-standard, method, denoted using a subscript is used for problems with BA, piecewise-constant This method uses the average of between nodes and as D(x). D(x) the value of at the control volume boundary between those nodes, i.e. D
Appendix D: Details of FVM implementations of diffusion term This section details the methods used to approximate the diffusion coefficient at the control volume boundaries when the diffusion coefficient is piecewise constant and either over the entire domain or known only at the nodes. Average of node value This standard method uses simple linear interpolation to estimate the diffusion coefficient at the control volume boundaries using the values of D at the adjacent nodes [9-10]. It is implemented here in situations where D is only known at the nodes but is also used where D is piecewise constant. In the piecewise-constant case, it makes no use of the fact that D(x) is known between the nodes. The general equation for the diffusion coefficient at the boundary between nodes n and n+1 is the superscript An refers to “average of the node values”) which, for uniform node spacing, simplifies to Average of average values The second non-standard approach denoted using a superscript AA, and only used where D is piecewise-constant, takes account of the diffusion coefficient at all points by first calculating the average values of D over each volume using The diffusion coefficient at a control volume boundary is then calculated as Average between the nodes The non-standard, method, denoted using a subscript BA, is used for problems with piecewise-constant D(x). This method uses the average of D(x) between nodes and as the value of D at the control volume boundary between those nodes, i.e.
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