Consider a simple model of a problem with constant coefficients = 1, = 2, = 120, D K v and = 30 modelled using = 1.0×10−6 and = 5 over the domain with Dirichlet S t N boundary conditions = 0 and = 1 and the initial condition = 0, 1. V(0,t) V(1,t) V(x,0) x ≠ The Peclet number is 30. shows solutions at = 0.001 and = It can be observed that t t ∞. 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 below which the Pe 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 (which equals under Rn Rn−1 those conditions) and 1/Rn + 1/Rn−1 + + to be written as Gl,n Gr,n−1
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
Numerical_modelling_of_convection-reaction-diffusion_problems_using_electrical_analogues.docx