The values of c in Eq. will depend on the problem coefficients and solution. They will also depend on h. In general, they will change monotonically as h is decreased, approaching constant values as h approaches zero. As a result, the error in any solution value will, as h approaches zero, always reduce when h is reduced in a manner that is consistent with the order of the scheme. For example, with a second-order accurate scheme, the error in any solution value will reduce by a factor of approximately four when the node spacing is halved. For some types of problems, numerical schemes may not be “consistently convergent”, i.e. the error at a point in the solution may not always decrease as h is decreased, even as h approaches zero, or may decrease as h is decreased, but not in a way that is consistent with the order of the method. The value of p for a given scheme can be estimated empirically in a number of ways, giving an estimated order of convergence (EOC). If an exact solution is known then it can be estimated as [80] where eh is the error in the solution obtained using a step size h [81]. When an exact solution is not available, the EOC can be estimated using where Vh , Vh/2 and Vh/4 are corresponding solution values estimated using step sizes of h, h/2 and h/4, respectively [80]. If the results used to estimate the order of convergence are obtained using a scheme that is consistently convergent for the problem tested, then, as h approaches zero, and assuming that round-off errors in the results are insignificant, the estimated order of convergence should approach the order of accuracy of the scheme (e.g. for a scheme, the value of EOC should approach two as h is decreased). Benchmark numerical solutions A benchmark numerical solution is simply a highly accurate numerical approximation that can be used to estimate errors in less accurate estimates, and to compare the
Breast with embedded tumour Breast thermography is used as a complementary examination method for breast cancer identification. It is a painless procedure involving the use of infrared imaging to measure skin temperature. for a breast with a developing tumour, this temperature can be higher than normal by 1 to 3 degrees, it can be used to detect such tumours [26, 100-101]. Numerical modelling has been used to understand and help predict how tumours of different sizes and at different locations affect the skin temperature. The model considered here is similar to that examined by Amri et al.[26]. The governing equation used to model the heat transfer in a breast is, when simplified to one dimension, where , c and k are the tissue density, heat capacity and thermal conductivity, b is the blood perfusion rate, cb is the blood heat capacity, Qm is metabolic heat generation rate and Ta is the arterial blood temperature. In the one-dimensional transient study conducted by Amri et al, the surface of the breast is located at x = 0 and the tumour is located at a distance of x = . The surface of the breast is assumed to be at a constant temperature, T∞, which could be produced through contact with a hot or cold plate. The temperature of the tumour is also assumed to be constant, Tc. The model is illustrated in . Figure 32: One-dimensional of the healthy tissue between tumour and breast surface. The initial temperature is given by [26] where A = bbcb/k. The results presented here are for a problem with model length  = 50mm, tissue properties of ρ = ρb = 1050kg/m3, c = cc = 3800J/kg°C, h0 = 20W/m2 , ωb =, Qm = 400W/m3, and k = 0.5W/m°C, and with temperatures of Tc = Ta= 37°C, Tf = 28°C and T∞ = 49°C. The temperature profile between the skin and the tumour has
Consider a simple model of a problem with constant coefficients D = 1, K = 2, v = 120, and S = 30 modelled using t = 1.0×10−6 and N = 5 over the domain with Dirichlet boundary conditions V(0,t) = 0 and V(1,t) = 1 and the initial condition V(x,0) = 0, x ≠ 1. The Peclet number is 30. shows solutions at t = 0.001 and t = ∞. It can be observed that the QUICK FVM solution exhibits wiggles in both steady state and transient conditions. The steady-state LCM solution is exact, while the transient solution contains both spatial and temporal errors. Physically, the values of the solution of a convection-diffusion problem should be bounded by the boundary values (e.g. all solution values, in this case, should be between 0 and 1). This is not the case if the scheme exhibits wiggles. One method of determining, for a particular scheme, the limit on the value of Pe below which the solution will be bounded, is Godunov’s theorem [47]. If a reaction term is included in the problem with a positive reaction coefficient, then the solution will no longer always be bounded by the boundary values, but, if those boundary values are both positive, then all solution values must also be positive (i.e. the solution values must be bounded by zero and the larger of the two boundary values). Consider the general discretized LCM equation for convection-reaction-diffusion equation without a source term and with coefficients in steady state as given in Eq. . Rearranging this equation gives (a) (b) Figure 42: Transient, (a), and (b), LCM and QUICK FVM solutions for a problem with constant coefficients. Now combining Equations to and Equations to allows Rn (which equals Rn−1 under those conditions) and 1/Rn + 1/Rn−1 + Gl,n + Gr,n−1 to be written as
which, when differentiated with respect to x, gives Replacing Ix using Eq. and rearranging the resulting equation gives Similarly, for the y-direction, Now, according to Kirchhoff’s current law, the sum of the currents entering the junction of the two TLs must equal zero, and so which simplifies to Using Equations and to replace the derivatives of the currents gives From Eq. , rx = y, and so this can be rewritten as Now, from Eq. , , and so this becomes It is clear that a transmission line network of this kind is analogous to the CRDE in two dimensions when
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.