An accurate and efficient Chebyshev expansion method for large-scale transient heat conduction problems

High-order implicit time integration scheme based on Padé expansions

“…Similar to the discussion on the HHT-α method, the relative period errors and amplitude ratios are shown in Fig. 10 for orders (1, 2), (2, 3), (3,4) and (4, 5). To explain the observations and conclusions concisely, the order of the high-order scheme is denoted as (L = M − 1, M ).…”

“…The relative period errors given by Eq. (72) for orders (1, 2), (2, 3), (3,4) and (4, 5) of the mixed Padé scheme are plotted in Fig. 7, together with those of HHT-α scheme.…”

“…In this section, the time step size for the order (1, 2) scheme is chosen to have a CFL number of 10 leading to ∆t = 9.8658 × 10 −6 , which means in each time step the wave travels through 10 linear elements. Following the discussions in Section 5.5.2, the time step sizes of orders (2, 3), (3,4) and (4, 5) are chosen by multiplying that of order…”

“…(1, 2) with a factor of 2, 3 and 4, respectively, to introduce a similar amount of numerical dissipation. These time step sizes correspond to CFL numbers of 20 for order (2, 3), 30 for order (3,4), and 40 for order (4, 5). In the subsequent analyses, the velocity at the middle point of the rod will be examined.…”

“…In several engineering and science disciplines, it is needed to evaluate the solution of the response history of a system to a dynamic action [1][2][3]. Various direct time integration methods have been developed to discretize the time-continuous dynamic equations [4,5].…”