TLM as a time-stepping method for FVM Consider the FVM approximation of a reaction-diffusion equation When compared with the equivalent equation for LCM it is that a lumped-component circuit with parameters , is analogous to the FVM approximation. Solving the equations for the voltage at nodes in such a circuit is the same as solving the FVM equations. As with LCM, the capacitors can be replaced with TL segments and the time can be performed using TLM as described above. Testing of the TLM time-stepping scheme for LCM and FVM To test the two schemes, a problem involving heat conduction in a plate with constant coefficients is considered here. The domain consists of three physical layers with piecewise-constant coefficients = {0.02,0.08,0.05}cm2/s, = {0.04,0.6,0.14} D S °C/s and = {0.2,0.4,0.6}s−1, as shown in . The node spacing is = 0.1 cm with K h discontinuities at = 0.3 cm and = 0.6 cm. The boundary conditions (i.e. the x x temperatures at the two faces of the plate) are = 0 and = 1, and the initial V(0,t) V(1,t) conditions are = 0. V(x,0)
TLM as a time-stepping method for FVM Consider the FVM approximation of a reaction-diffusion equation When compared with the equivalent equation for LCM it is that a lumped-component circuit with parameters , is analogous to the FVM approximation. Solving the equations for the voltage at nodes in such a circuit is the same as solving the FVM equations. As with LCM, the capacitors can be replaced with TL segments and the time can be performed using TLM as described above. Testing of the TLM time-stepping scheme for LCM and FVM To test the two schemes, a problem involving heat conduction in a plate with coefficients is considered here. The domain consists of three physical layers with piecewise-constant coefficients D = {0.02,0.08,0.05}cm2/s, S = {0.04,0.6,0.14} °C/s and K = {0.2,0.4,0.6}s−1, as shown in . The node spacing is h = 0.1 cm with discontinuities at x = 0.3 cm and x = 0.6 cm. The boundary conditions (i.e. the temperatures at the two faces of the plate) are V(0,t) = 0 and V(1,t) = 1, and the initial conditions are V(x,0) = 0. constant