Multiscale modeling is playing an important role in the rapid advance of many areas of engineering and science, including nanotechnology, materials, cell and molecular biology, and geological and earth sciences. The efficient simulation of these problems can be quite challenging, due to the widely separated time and length scales which are often strongly coupled. It is important and necessary to develop simulation methods that can accurately and efficiently capture multiscale and multiphenomena features.

In the first part of this work we study a class of stochastic methods, namely Monte Carlo methods, with applications to the formation of supported lipid bilayers and copper electrodeposition. A hybrid multiscale kinetic Monte Carlo (HMKMC) method for efficient and accurate simulation of copper electrodeposition is developed. The fast diffusion events are simulated deterministically with a heterogeneous diffusion model that considers the site-blocking effects of additives. The chemical reactions are simulated by an adaptive, non-negativity preserving tau-leaping stochastic simulation algorithm (SSA), where the stepsize is selected automatically for best speed-up while retaining a desired accuracy. Numerical results indicate that the new HMKMC method is accurate and highly efficient.

In the second part, we develop an explicit stabilized Runge-Kutta-Chebyshev projection (RKCP) method for solving differential-algebraic equations (DAEs), with applications to the solution of incompressible Navier-Stokes equations and the Lagrange multiplier based domain decomposition method. The RKCP method requires only one projection per step, independent of the number of Runge-Kutta stages. An additional projection is performed whenever a second-order approximation for the pressure is desired. We further generalize the projection ideas and propose a framework for analysis of second-order projection methods. Based on this framework, we perform a full analysis of accuracy and stability for the second-order projection methods. The RKCP method is also proposed to solve parabolic PDEs with a Lagrange multiplier based domain decomposition method. With mass lumping techniques and node-to-node matching grids, this method is fully explicit without solving a linear system, and straightforward to implement and to parallelize. Numerical results show that the method has excellent performance.