Module method_of_characteristics 

Method of characteristics for first-order PDEs

Implements the method of characteristics to solve first-order quasi-linear PDEs of the form:

a(x,y,u)∂u/∂x + b(x,y,u)∂u/∂y = c(x,y,u)

Mathematical Background

The method of characteristics transforms a first-order PDE into a system of ordinary differential equations (characteristic equations). The solution follows characteristic curves in the (x, y, u) space.

Characteristic Equations:

dx/ds = a(x,y,u)
dy/ds = b(x,y,u)
du/ds = c(x,y,u)

where s is a parameter along the characteristic curve.

Algorithm:

1. Extract coefficients a, b, c from PDE
2. Construct characteristic ODE system
3. Solve ODEs using numerical integration (RK4)
4. Eliminate parameter s to get implicit solution

Domain and Limitations

• Supported: First-order quasi-linear PDEs with two independent variables
• Requires: Coefficients a, b, c must be continuous
• Singularities: Division by zero in coefficients is detected and rejected

References

• Evans, L. C. (2010). Partial Differential Equations. AMS. Chapter 3.
• Logan, J. D. (2015). Applied Partial Differential Equations. Springer. Chapter 2.

Structs

CharacteristicSolution

Result of applying the method of characteristics

PdeCoefficients

Coefficients of a first-order quasi-linear PDE

Enums

CharacteristicsError

Error type for method of characteristics

Functions

method_of_characteristics

Applies the method of characteristics to a first-order quasi-linear PDE

solve_characteristic_odes

Solve characteristic ODE system numerically