Overview of Other Numerical Methods

advective (convective) Finite Volume Methods Finite difference methods are based on a discretization of the differential form of the conservation equations. Finite volume methods are based on a discretization of the integral forms of the conservation equations: advective (convective) fluxes other transports (diffusion, taxis, etc) sum of sources and sinks (reactions, etc) “Finite volume” refers to the small volume surrounding each node point on a mesh.

Finite Volume Methods In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. These give 2 advantages of finite volume: Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. It is easily formulated to allow for unstructured meshes (arbitrary polyhedra in 3D or polygons in 2D).

Spectral Methods Spectral methods involve the use of Fast Fourier Transform (FFT). The idea is to write the solution of the differential equation as a sum of certain “basis functions” (for example, as a Fourier series which is a sum of sinusoids) and then to choose the coefficients in the sum in order to satisfy the differential equation as well as possible. Spectral methods and finite element methods are built on the same idea. The main difference is that spectral methods take on a global approach while finite element methods use a local approach.

Other Methods Particle methods are mesh-free methods that are used to trace the dynamics of singularities and approximate diffusive and dispersive phenomena. The particle-in-cell (PIC) method was advocated in computational plasma physics. The class of vortex methods are used to simulate incompressible and slightly compressible flows. The immersed boundary method, which originated in the context of bioflows, traces the time evolution of the boundary of elastic structures which are immersed in incompressible fluids.

Other Methods The level-set method is used for tracking interfaces and shapes by realizing them in as the zero-level set of higher-dimensional smooth surfaces. In the front tracking method, a separate grid is used to mark and trace the interface whereas a fixed grid is used in between these interfaces. Wavefront methods and the moment method are encountered in high-frequency computations, offering an alternative to traditional geometrical optics techniques based on ray tracing.

Individual Cell-Based Models Generally, individual cell-based models are classified into lattice-based and off-lattice models. In the lattice-based modeling, each biological cell is represented as either a single lattice site or a set of many contiguous sites on a lattice through spatial discretization. Models that fall into the category of lattice-based modeling generally use approaches that include cellular automata models, some hybrid discrete-continuum or HDC (combination between continuum and discrete cellular automata like) models, Glazier-Graner-Hogeweg or GGH models, and the invariants and extensions of these models.

Individual Cell-Based Models On the other hand, in the off-lattice modeling each biological cell is often treated as a unit of finite volume with arbitrary locations whose motion is not restricted to lattice points and has shapes that are restricted to spheres, ellipsoids, or Delaunay-decomposition-based shapes (or Voronoi polygons or Thiessen polygons)

Individual Cell-Based Models Difference between lattice-based and off-lattice methods:

References Wikipedia Video 1 (Slide 10): Modeling the Influence of the E-Cadherin-beta-Catenin Pathway in Cancer Cell Invasion: A Multiscale Approach (2008) I. Ramis-Conde, D. Drasdo, A.R.A. Anderson, M.A.J. Chaplain, Biophysical Journal 95(1):155-165. Video 2 (Slide 11) & Figure (Slide 9): Integrating Intercellular Dynamics Using CompuCell3D and Bionetsolver: Applications to Multiscale Modelling of Cancer Cell Growth and Invasion (2012) V. Andasari, R.T. Roper, M.H. Swat, M.A.J. Chaplain, PLoS ONE 7(3):e33726. Video 3 (Slide 12): 3D Multi-Cell Simulation of Tumor Growth and Angiogenesis (2009) A. Shirinifard, J.S. Gens, B.L. Zaitlen, N.J. Poplawski, M.H. Swat, J.A. Glazier, PLoS ONE 4(10):e7190.