Background Info
Quick definitions of variables and concepts used in the solver interface.
Concepts
Explicit method updates the next time step using only known values from the current step. It is simple but requires a small time step for stability.
Implicit method solves a linear system at each time step. It is stable for larger steps but more computationally intensive.
Crank-Nicolson (CN) averages explicit and implicit updates for second-order accuracy in time, but can show oscillations near discontinuities.
Rannacher stepping replaces the first CN step with multiple implicit substeps to damp early-time oscillations.
Black-Scholes is a PDE model for European option prices that balances diffusion (volatility) and drift (risk-free rate).
Heat Equation Variables
- x: spatial variable along the 1D domain.
- t: time variable.
- a: left boundary of the spatial domain.
- b: right boundary of the spatial domain.
- l: final time (time horizon).
- n: number of spatial steps (grid resolution in x).
- m: number of time steps.
- f(x,t): forcing term in the PDE.
- u(x,0): initial temperature profile.
- u(a,t): boundary value at the left endpoint.
- u(b,t): boundary value at the right endpoint.
- h: spatial step size, h = (b - a) / n.
- k: time step size, k = l / m.
- r: stability ratio, r = k / h^2 (for explicit heat scheme).
Black-Scholes Variables
- K: strike price of the option.
- T: time to maturity.
- r: risk-free interest rate (annualized).
- sigma: volatility (annualized).
- s: asset price variable in the PDE.
- s_max: truncation boundary, typically 4K.