Note: This video has been updated. I found the reason for the discrepancy between research code and commercial software: the grid was finer in the latter... The plot of 2D grid has been replaced, because it had 160x160 instead of 80x80 cells. Replaced is also the plot in slide 20.
In this lecture the performance of some of the time-integration methods described in the two previous two lectures is analyzed on two test cases. The first test case is the unsteady version of the one-dimensional problem studied in Lecture 6, with one difference: now one of the boundary values varies sinusoidally in time. Although of little practical relevance, the test case is still useful for testing the features of time-integration methods. Iteration errors are minimized to round-off level because an efficient direct solver for one-dimensional problems is used - the TDMA algorithm described in Lecture 10. Nine schemes were implemented in the test code , which is available in supplementary material for this lecture: 1st-order explicit and implicit Euler methods, 2nd-order Crank-Nicolson, quadratic backward, Adams-Bashforth and Runge-Kutta methods, and 4th-order Adams-Bashforth, Adams-Moulton, and Runge-Kutta methods. The second test case is the unsteady version of the problem studied in Lecture 9. Here only 1st and 2nd-order methods are analyzed. Only the transport equation for a scalar variable is solved - the velocity field is prescribed analytically and is steady. The initial condition is a zero scalar field, while a linear variation of the scalar variable is prescribed on one boundary and we are observing the propagation of the boundary influence into the field due to both convection and diffusion transport.
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