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Created readme

Author: xtalax

README.md

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+# MethodOfLines.jl
+
+Provides automatic discretization of symbolic PDE systems as defined with ModelingToolkit.jl
+
+Usage:
+```
+discretization = MOLFiniteDifference(dxs, 
+                                      <your choice of continuous variable, usually time>; 
+                                      upwind_order = <currently hard coded to 1>, 
+                                      centered_order = <currently hard coded to 2>, 
+                                      grid_align = your grid type choice>)
+prob = discretize(pdesys, discretization)
+```
+Where dxs is a vector of pairs of parameters to the grid step in this dimension, i.e. `[x=>0.2, y=>0.1]`
+
+Currently supported grid types: `center_align` and `edge_align`
+center_align: naive grid, starting from lower boundary, ending on upper boundary with step of dx
+edge_align: offset grid, set halfway between the points that would be generated with center_align, with extra points at either end that are above and below the supremum and infimum by dx/2.
+This provides higher accuracy with neumann/robin boundary conditions.
+
+Currently boundary conditions defined in terms of derivatives at the boundary are unsupported.
+
+## Full Example:
+```
+## 2D Diffusion
+
+# Variables, parameters, and derivatives
+@parameters t x y
+@variables u(..)
+Dxx = Differential(x)^2
+Dyy = Differential(y)^2
+Dt = Differential(t)
+# Domain edges
+t_min= 0.
+t_max = 2.0
+x_min = 0.
+x_max = 2.
+y_min = 0.
+y_max = 2.
+
+# Discretization parameters
+dx = 0.1; dy = 0.2
+order = 2
+
+# Analytic solution for boundary conditions
+analytic_sol_func(t, x, y) = exp(x + y) * cos(x + y + 4t)
+
+# Equation
+eq  = Dt(u(t, x, y)) ~ Dxx(u(t, x, y)) + Dyy(u(t, x, y))
+
+# Initial and boundary conditions
+bcs = [u(t_min,x,y) ~ analytic_sol_func(t_min, x, y),
+        u(t, x_min, y) ~ analytic_sol_func(t, x_min, y),
+        u(t, x_max, y) ~ analytic_sol_func(t, x_max, y),
+        u(t, x,y_min) ~ analytic_sol_func(t, x, y_min),
+        u(t, x, y_max) ~ analytic_sol_func(t, x, y_max)]
+
+# Space and time domains
+domains = [t ∈ Interval(t_min, t_max),
+           x ∈ Interval(x_min, x_max),
+           y ∈ Interval(y_min, y_max)]
+
+# PDE system
+@named pdesys = PDESystem([eq], bcs, domains, [t, x, y], [u(t, x, y)])
+
+# Method of lines discretization
+discretization = MOLFiniteDifference([x=>dx,y=>dy],t;centered_order=order)
+prob = ModelingToolkit.discretize(pdesys,discretization)
+
+# Solution of the ODE system
+sol = solve(prob,Tsit5())
+```