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1. Add in second spatial dimension $$\frac{\partial u }{\partial t} = \epsilon \left(\frac{\partial^2 }{\partial y^2}+ \frac{\partial^2 }{\partial x^2} \right)u+ uf(u) - vg(u), \quad \frac{\partial v }{\partial t}= \left(\frac{\partial^2 }{\partial x^2}+ \frac{\partial^2 }{\partial y^2} \right) v + vh(u). $$ This modifies the equation interface and equation builder subroutines
No due date1. Modify the heat equation to solve $$\frac{\partial u }{\partial t} = \frac{\partial^2 u}{\partial x^2},\quad \frac{\partial v }{\partial t}= \frac{\partial^2 v}{\partial x^2} $$ as a coupled system. 2. Include true coupling where the two equations are different. $$\frac{\partial u }{\partial t} = \frac{\partial^2 u}{\partial x^2} +u+v ,\quad \frac{\partial v }{\partial t}= \frac{\partial^2 v}{\partial x^2}+v+u$$ as a coupled system. Solutions should be identical as a test.
No due dateAdd in a temporal marching scheme into the exist code to solve problems with a temporal derivative. None of the coefficients of the equation need to explicitly depend in time. https://computing.llnl.gov/projects/odepack, https://people.sc.fsu.edu/~jburkardt/f77_src/odepack/odepack.html ??? Can discuss von Neumann stability conditions (i.e. the step size requirements in $x$ and $t$) - wonder how the adapative marching deals with this?
No due dateModify the existing code to solve a singular pertubation problem of the form: $$\epsilon u'' + (1+\epsilon)u' + u = 0,\quad u(0) = 0, u(1) = 1, 0<\epsilon<<0 $$ Via a streched grid at the boundary x = 0.
No due date•3/3 issues closedBuild a boundary value solver using finite differences: 1D problem - $ u_xx = u, u'(0) = 0, u(1)=0 $ for example - Discretise using finite differences + compressed storage (or Chebyshev polynomials) - will hopefully use an external package to help with this! (Any tips?) - Then use some linear algebra package that works with FPM to solve the problem. - Compare with analytic solutions This is actually completed - however it's all trapped inside a submodule I can't get out!
No due date•4/4 issues closed