In this thesis, some advanced numerical schemes are developed for applications in modeling transient non-continuum heat conduction and surface diffusion motion to improve the accuracy and efficiency.

In the first part, a new set of modified ballistic-diffusive equations are proposed for the modeling of transient non-continuum heat conduction. For non-continuum phonon transport, the phonon Boltzmann transport equation is regarded as an accurate model when the wave effect is negligible. However it is difficult and computationally intensive to solve. The ballistic-diffusive approach (BDE) proposed by Chen [G. Chen, phys. Rev. Lett. 86, 2297 (2001)] greatly simplifies the solution procedure and reduces the computational cost by modeling the ballistic and diffusive parts separately and usin...[ Read more ]

In this thesis, some advanced numerical schemes are developed for applications in modeling transient non-continuum heat conduction and surface diffusion motion to improve the accuracy and efficiency.

In the first part, a new set of modified ballistic-diffusive equations are proposed for the modeling of transient non-continuum heat conduction. For non-continuum phonon transport, the phonon Boltzmann transport equation is regarded as an accurate model when the wave effect is negligible. However it is difficult and computationally intensive to solve. The ballistic-diffusive approach (BDE) proposed by Chen [G. Chen, phys. Rev. Lett. 86, 2297 (2001)] greatly simplifies the solution procedure and reduces the computational cost by modeling the ballistic and diffusive parts separately and using the first-order spherical harmonic function to approximate the diffusive intensity.

The accuracy of the BDE remains to be improved particularly near the boundary and at large time scales. In this part, a deficiency in the BDE is identified, and a new physically sound boundary model is proposed, which leads to a new set of ballistic-diffusive equations. Several benchmark problems are employed to validate the performance of the modified ballistic-diffusive approach, and results indicate the new approach is much more accurate than the original BDE and yet retains the same level of efficiency. Hence it is an effective method for the modeling of transient non-continuum heat conduction.

In the second part, a high-order level set method based on the total variation diminishing Runge-Kutta method, a high-order scheme for distance computation and a smoothing scheme are developed for simulating curvature driven flow and surface diffusion motion, which overcomes the high-order CFL time restriction. The enhanced stability is achieved by utilizing several techniques, resulting in an accurate and smooth velocity field. In particular, the scheme for distance computation is used to reinitialize the level-set function and to extend the velocity from the zero level-set to the remaining of the domain. As such, it greatly reduces the accumulated errors typically observed in the traditional PDE-based methods. The smoothing technique is used to remove the high-frequency oscillations procedure by the high-order derivatives of the level-set function and is the key to overcome the CFL restriction on the time step. Results on several benchmark problems have demonstrated that compared with the semi-implicit methods, the developed method is more accurate and achieves the same, if not better, stability.

Although the level set method can handle topological changes naturally, its performance in simulating curvature driven flows and surface diffusion motions highly depends on how accurate the interface curvature and intrinsic laplacian of curvature can be simulated, particularly in cases when topological change happens at interfaces that are close by. Based on the high-order accurate approach developed by Saye for computing the minimum distance to implicitly defined surfaces, we developed a local treatment that can handle general interface configurations including multiple interfaces that are close by. Its performances in calculating curvature and surface diffusion motion are tested and good results are obtained.

Keywords: Non-continuum phonon transport; Boltzmann transport equation; ballistic-diffusive Approximation; level set method; curvature flow; surface diffusion; spline smooth; high order closest point calculation; CFL restriction.