Three novel truly-explicit time-marching procedures considering adaptive dissipation control

Abstract

In this paper, three novel truly-explicit time-marching techniques are proposed for the solution of hyperbolic models. In these techniques, locally computed time-integration parameters are applied, which are defined taking into account the properties of the spatially/temporally discretised model and the evolution of the computed responses. These adaptive time integrators allow introducing enhanced numerical damping into the analysis, reducing spurious non-physical oscillations that occur due to the excitation of spatially unresolved modes. The proposed adaptive techniques are formulated based on second-, third-, and fourth-order accurate time-marching approaches, providing solution algorithms with different complexities. At the end of the paper, numerical results are presented, illustrating the good performance of the proposed procedures.

Appendix

The entries of the amplification matrices A2, A3 and A4 may be specified as follows:

$$A2_{11} =1 - \tfrac{1}{2}w^{2} \Delta t^{2} + \tfrac{1}{2}\,\xi {\kern 1pt} w^{3} \Delta t^{3} .$$

(A1a)

$$A2_{12} =\Delta t - \xi w\Delta t^{2} - (\tfrac{1}{4}\alpha - \xi^{2} )w^{2} \Delta t^{3} + \tfrac{1}{4}\alpha \xi w^{3} \Delta t^{4} .$$

(A1b)

$$A2_{21} = - w^{2} \Delta t + \xi {\kern 1pt} w^{3} \Delta t^{2} .$$

(A1c)

$$A2_{22} =1\, - 2\xi {\kern 1pt} w\Delta t - (\tfrac{1}{2}\alpha - 2\xi^{2} ){\kern 1pt} w^{2} \Delta t^{2} + \tfrac{1}{2}\alpha \,\xi {\kern 1pt} w^{3} \Delta t^{3} .$$

(A1d)

$$A3_{11} =1 - \tfrac{1}{2}w^{2} \Delta t^{2} + \tfrac{1}{3}\xi {\kern 1pt} w^{3} \Delta t^{3} + \tfrac{1}{24}{\kern 1pt} w^{4} \Delta t^{4} .$$

(A2a)

$$A3_{12} =\Delta t - \xi {\kern 1pt} w\Delta t^{2} - (\tfrac{1}{12} + \tfrac{1}{12}\alpha - \tfrac{2}{3}\xi^{2} )w^{2} \Delta t^{3} + (\tfrac{1}{12} + \tfrac{1}{18}\alpha )\,\xi {\kern 1pt} w^{3} \Delta t^{4} + \tfrac{1}{144}\alpha w^{4} \Delta t^{5} .$$

(A2b)

$$A3_{21} = - w^{2} \Delta t + \xi {\kern 1pt} w^{3} \Delta t^{2} + (\tfrac{1}{6} - \tfrac{2}{3}\xi^{2} )w^{4} \Delta t^{3} - \tfrac{1}{12}\xi {\kern 1pt} w^{5} \Delta t^{4} .$$

(A2c)

$$\begin{gathered} A3_{22} =1\, - 2\xi {\kern 1pt} w\Delta t - (\tfrac{1}{3} + \tfrac{1}{6}\alpha - 2\xi^{2} ){\kern 1pt} w^{2} \Delta t^{2} + (\tfrac{1}{2} + \tfrac{1}{6}\alpha - \tfrac{4}{3}\xi^{2} )\,\xi {\kern 1pt} w^{3} \Delta t^{3} + \hfill \\ \quad \quad \quad (\tfrac{1}{36}\alpha - (\tfrac{1}{6} + \tfrac{1}{9}\alpha )\xi^{2} )w^{4} \Delta t^{4} - \tfrac{1}{72}\alpha \xi w^{5} \Delta t^{5} . \hfill \\ \end{gathered}$$

(A2d)

$$A4_{11} =1 - \tfrac{1}{2}w^{2} \Delta t^{2} + \tfrac{1}{3}\xi {\kern 1pt} w^{3} \Delta t^{3} + (\tfrac{1}{24} - \tfrac{1}{6}\xi^{2} ){\kern 1pt} w^{4} \Delta t^{4} - \tfrac{23}{{720}}\,\xi {\kern 1pt} w^{5} \Delta t^{5} - \tfrac{1}{720}w^{6} \Delta t^{6} .$$

(A3a)

$$\begin{gathered} A4_{12} =\Delta t - \xi {\kern 1pt} w\Delta t^{2} - (\tfrac{1}{12} + \tfrac{1}{12}\alpha - \tfrac{2}{3}\xi^{2} )w^{2} \Delta t^{3} + (\tfrac{1}{9} + \tfrac{1}{18}\alpha - \tfrac{1}{3}\xi^{2} )\,\xi {\kern 1pt} w^{3} \Delta t^{4} \hfill \\ \quad \quad \quad +(\tfrac{1}{288} + \tfrac{1}{144}\alpha - (\tfrac{23}{{360}} + \tfrac{1}{36}\alpha )\xi^{2} )\,w^{4} \Delta t^{5} - (\tfrac{1}{360} + \tfrac{23}{{4320}}\alpha )\,\xi {\kern 1pt} w^{5} \Delta t^{6} - \tfrac{1}{4320}\alpha \,{\kern 1pt} w^{6} \Delta t^{7} . \hfill \\ \end{gathered}$$

(A3b)

$$\begin{gathered} A4_{21} = - w^{2} \Delta t + \xi {\kern 1pt} w^{3} \Delta t^{2} + (\tfrac{1}{6} - \tfrac{2}{3}\xi^{2} )w^{4} \Delta t^{3} - (\tfrac{1}{6} - \tfrac{1}{3}\xi^{2} )\,\xi {\kern 1pt} w^{5} \Delta t^{4} - \hfill \\ \quad \quad \quad (\tfrac{1}{180} - \tfrac{23}{{360}}\xi^{2} )w^{6} \Delta t^{5} + \tfrac{1}{360}\,\xi {\kern 1pt} w^{7} \Delta t^{6} . \hfill \\ \end{gathered}$$

(A3c)

$$\begin{gathered} A4_{22} =1\, - 2\xi {\kern 1pt} w\Delta t - (\tfrac{1}{3} + \tfrac{1}{6}\alpha - 2\xi^{2} ){\kern 1pt} w^{2} \Delta t^{2} + (\tfrac{1}{2} + \tfrac{1}{6}\alpha - \tfrac{4}{3}\xi^{2} )\,\xi {\kern 1pt} w^{3} \Delta t^{3} \hfill \\ \quad \quad \quad +(\tfrac{1}{72} + \tfrac{1}{36}\alpha - (\tfrac{7}{18} + \tfrac{1}{9}\alpha )\xi^{2} + \tfrac{2}{3}\xi^{4} )\,w^{4} \Delta t^{4} - (\tfrac{13}{{720}} + \tfrac{1}{36}\alpha - (\tfrac{23}{{180}} + \tfrac{1}{18}\alpha )\xi^{2} )\,\xi {\kern 1pt} w^{5} \Delta t^{5} \hfill \\ \quad \quad \quad -(\tfrac{1}{1080}\alpha - (\tfrac{1}{180} + \tfrac{23}{{2160}}\alpha )\xi^{2} )\,w^{6} \Delta t^{6} + \tfrac{1}{2160}\alpha \xi {\kern 1pt} w^{7} \Delta t^{7} . \hfill \\ \end{gathered}$$

(A3d)

whereas the entries related to the load operator vectors L2, L3 and L4 may be expressed as:

$$L2_{11} =\tfrac{1}{2}\Delta t\,(1 - \xi w\Delta t).$$

(A4a)

$$L2_{21} =1 - \xi w\Delta t.$$

(A4b)

$$L2_{12} =0.$$

(A4c)

$$L2_{22} =0.$$

(A4d)

$$L3_{11} =\tfrac{1}{24}\Delta t\,(12 - 8\xi w\Delta t - w^{2} \Delta t^{2} ).$$

(A5a)

$$L3_{21} =1 - \xi w\Delta t - \tfrac{1}{6}w^{2} \Delta t^{2} (1 - 4\xi^{2} ) + \tfrac{1}{12}\xi w^{3} \Delta t^{3} .$$

(A5b)

$$L3_{12} =\tfrac{1}{144}\Delta t^{2} \,(12 - 12\xi w\Delta t + 8\xi^{2} w^{2} \Delta t^{2} + \xi w^{3} \Delta t^{3} ).$$

(A5c)

$$L3_{22} = - \tfrac{1}{6}\xi w\Delta t^{2} + \tfrac{1}{6}\xi^{2} w^{2} \Delta t^{3} + \tfrac{1}{36}\xi w^{3} \Delta t^{4} (1 - 4\xi^{2} ) - \tfrac{1}{72}\xi^{2} w^{4} \Delta t^{5} .$$

(A5d)

$$L4_{11} =\tfrac{1}{720}\Delta t\,(360 - 240\xi w\Delta t - 30w^{2} \Delta t^{2} (1 - 4\xi^{2} ) + 23\xi w^{3} \Delta t^{3} + w^{4} \Delta t^{4} ).$$

(A6a)

$$\begin{gathered} L4_{21} =1 - \xi w\Delta t - \tfrac{1}{6}w^{2} \Delta t^{2} (1 - 4\xi^{2} ) + \tfrac{1}{6}\xi w^{3} \Delta t^{3} (1 - 2\xi^{2} ) \hfill \\ \quad \quad \quad + \tfrac{1}{360}w^{4} \Delta t^{4} (2 - 23\xi^{2} ) - \tfrac{1}{360}\xi w^{5} \Delta t^{5} . \hfill \\ \end{gathered}$$

(A6b)

$$\begin{gathered} L4_{12} =\tfrac{1}{34560}\Delta t^{2} \,(2880 - 2880\xi w\Delta t - 120w^{2} \Delta t^{2} (3 - 16\xi^{2} ) + 120\xi w^{3} \Delta t^{3} (3 - 8\xi^{2} ) + \hfill \\ \quad \quad \quad w^{4} \Delta t^{4} (15 - 184\xi^{2} ) - 8\xi w^{5} \Delta t^{5} ). \hfill \\ \end{gathered}$$

(A6c)

$$\begin{gathered} L4_{22} = - \tfrac{1}{6}\xi w\Delta t^{2} - \tfrac{1}{24}w^{2} \Delta t^{3} (1 - 4\xi^{2} ) + \tfrac{1}{9}\xi w^{3} \Delta t^{4} (\tfrac{7}{16} - \xi^{2} ) + \hfill \\ \quad \quad \quad \tfrac{1}{576}w^{4} \Delta t^{5} (1 - 20\xi^{2} + 32\xi^{4} ) - \xi w^{5} \Delta t^{6} (\tfrac{31}{{17280}} - \tfrac{23}{{2160}}\xi^{2} ) + \tfrac{1}{2160}\xi^{2} w^{6} \Delta t^{7} . \hfill \\ \end{gathered}$$

(A6d)