| Email: | | jhomenuk at physics.ubc.ca |
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Research: Finite-difference Approximations
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Honours Thesis: Preliminary Results
- Simple activator-inhibitor system: This attempts to model a hypothetical biological system in development. The components begin with nearly equal and uniform distributions. The activator is given a slight defect, causing the system to react in a preferred direction, but eventually form a stable periodic structure.
- Small Field [ MPEG ]
- Large Field [ MPEG ]
- (2+1) Diffusion Equation using Multigrid [ diffusion2D.sdf ]. The above activator-inhibitor system will be extended to (2+1) dimensions.
- Putting Multigrid to the test: [ MPEG ] This animation is the convergence of an elliptic equation to its solution sin(2 pi x). The initial guess is 2 sin(2 pi x). There is a 1-grid solution and a 2-grid solution. Each "time step" contains 10 iterations of the Gauss-Seidel method. For the 2-grid, 6 of those iterations are done on the coarse grid.
- (2+1) Reaction-Diffusion Equation: [ D15.0-a_0.sdf.1 | D15.0-h_0.sdf.1 ] A breakup instability in the weak interaction regime of the classical Gierer-Meinhardt model using homoclinic stripe initial conditions.
Solutions to Various Equations
| Description | n points | 2n points | 4n points | 8n points | 16n points | Convergence Test
Equality implies O(h2) convergence |
Wave Equation (1+1)
Cartesian Coordinates
Dirichlet/Sommerfeld BCs | MPEG | MPEG | MPEG | MPEG | MPEG | MPEG |
Wave Equation (1+1)
Spherical Coordinates
Neumann/Sommerfeld BCs | MPEG | MPEG | MPEG | MPEG | MPEG | MPEG |
Sine-Gordon Equation (1+1)
Spherical Coordinates
Neumann/Sommerfeld BCs | MPEG | MPEG | MPEG | MPEG | MPEG | MPEG |
KdV Equation (1+1)
- Cartesian coords: One Soliton [MPEG], Two Solitons [MPEG], convergence test [MPEG].
Klein-Gordon Equation with Double-Well Potential (1+1)
- Based on Ethan Honda's PhD dissertation [PS|PDF].
- Final Report [PS|PDF]
Reference Material
- RNPL Reference Manual [HTML|PS]
- RNPL User's Guide [HTML|PS]
- Robert Marsa's PhD dissertation, chapters 8 and 9 [PS]
- My essay [PDF] for RELG 100