PDE Quick Reference Card

Topic: advanced.pde.quick_reference

One-page cheat sheet for Method of Characteristics covering standard forms, solution templates, common patterns, shock formation, and troubleshooting guide. Includes code templates, decision trees, and physical applications.

Mathematical Definition

General Quasi-Linear Form: $$a(x,y,u)\frac{\partial u}{\partial x}+b(x,y,u)\frac{\partial u}{\partial y}=c(x,y,u)$$

Characteristic equations: $$\frac{dx}{ds}=a,\frac{dy}{ds}=b,\frac{du}{ds}=c$$

Transport Equation: $$\frac{\partial u}{\partial t}+c\frac{\partial u}{\partial x}=0$$ Solution: $u(x,t)=f(x-ct)$

Burgers' Equation: $$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=0$$ Solution: $u=f(\xi )$ where $x=\xi +f(\xi )t$ (implicit)

Shock speed (Rankine-Hugoniot): $$\frac{d{x}_{shock}}{dt}=\frac{[f(u)]}{[u]}=\frac{f(u_R)-f(u_L)}{u_R-u_L}$$

Entropy condition (Lax): $$f^{\prime }(u_L)>s>f^{\prime }(u_R)$$

PDE Quick Reference Card

One-page cheat sheet for Method of Characteristics

When to Use Method of Characteristics

Standard Forms

Transport Equation

$$\frac{\partial u}{\partial t}+c\frac{\partial u}{\partial x}=0$$

Solution: $u(x,t)=f(x-ct)$ where $f$ is initial condition

Physical meaning: Wave propagates right at speed $c$

Burgers' Equation (Nonlinear)

$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=0$$

Solution: $u=f(\xi )$ where $x=\xi +f(\xi )t$ (implicit)

Warning: Shocks can form! Use Rankine-Hugoniot for shock speed.

General Quasi-Linear Form

$$a(x,y,u)\frac{\partial u}{\partial x}+b(x,y,u)\frac{\partial u}{\partial y}=c(x,y,u)$$

Characteristic equations: $$\frac{dx}{ds}=a,\frac{dy}{ds}=b,\frac{du}{ds}=c$$

5-Step Solution Template

1. EXTRACT coefficients: a, b, c from PDE

2. BUILD characteristic system:
   dx/ds = a(x,y,u)
   dy/ds = b(x,y,u)
   du/ds = c(x,y,u)

3. SOLVE ODEs with IC: (x₀, y₀, u₀) = (ξ, 0, g(ξ))

4. ELIMINATE parameter s: solve for u(x,y)

5. VERIFY: Check PDE and IC satisfaction

Common Patterns

Pattern 1: Constant Coefficients

If $a$, $b$, $c$ are constants:

• Characteristics are straight lines
• Solution: $u=F(ax-by)$ where $F$ determined by IC

Pattern 2: Linear PDEs

If coefficients don't depend on $u$:

• Characteristics don't intersect
• Smooth solution exists globally

Pattern 3: Nonlinear PDEs

If coefficients depend on $u$:

• Characteristics can intersect → shocks form
• Use weak solutions + Rankine-Hugoniot + entropy condition

Shock Formation Checklist

When do shocks form?

1. ✅ PDE is nonlinear (coefficients depend on $u$)
2. ✅ Initial data has compression region ($\partial u_0/\partial x<0$)
3. ✅ Characteristics with different slopes intersect

Shock speed (Rankine-Hugoniot): $$\frac{d{x}_{shock}}{dt}=\frac{[f(u)]}{[u]}=\frac{f(u_R)-f(u_L)}{u_R-u_L}$$

Entropy condition (Lax): $$f^{\prime }(u_L)>s>f^{\prime }(u_R)$$ (Characteristics converge INTO shock)

Troubleshooting

Reference Formulas

Characteristic Equations

$$\frac{dx}{ds}=a,\frac{dy}{ds}=b,\frac{du}{ds}=c$$

General Solution Form

$$u(x,y)=F(I_1,I_2)$$ where $I_1$, $I_2$ are independent integrals of characteristic equations

Verification (PDE satisfaction)

$$a\frac{\partial u}{\partial x}+b\frac{\partial u}{\partial y}-c\stackrel{?}{=}0$$

Verification (IC satisfaction)

$$u(x,0)\stackrel{?}{=}g(x)$$

Physical Applications

Decision Tree: Which Method?

Is PDE first-order?
├─ YES → Is it quasi-linear?
│         ├─ YES → METHOD OF CHARACTERISTICS ✓
│         └─ NO → Specialized nonlinear techniques
└─ NO → What order?
          ├─ Second-order → Separation of variables, Fourier
          ├─ Higher-order → Advanced techniques
          └─ System → Vector method of characteristics

Performance Tips

1. Reuse PDE structures: Create Pde once, solve multiple times
2. Adjust ODE step size: Larger step = faster but less accurate
3. Simplify sparingly: Only when presenting results (expensive)
4. Parallel characteristics: Trace multiple characteristics in parallel

Common Mistakes (Avoid!)

❌ Wrong: Mixing up coefficient order ($a$ vs $b$) ✅ Right: Write PDE in standard form first

❌ Wrong: Forgetting to apply initial condition ✅ Right: General solution $u=F(\xi )$, IC determines $F$

❌ Wrong: Using Symbol::new("x") instead of macros ✅ Right: Always use symbol!(x) and expr!(...)

❌ Wrong: Ignoring shock formation in nonlinear PDEs ✅ Right: Check for characteristic intersection, apply shock theory

Print this page and keep it handy while solving PDEs!

Examples

Quick Code Template

Standard template for solving PDEs with method of characteristics

#![allow(unused)]
fn main() {
// Define symbols
let u = symbol!(u);
let t = symbol!(t);
let x = symbol!(x);

// Build PDE
let equation = expr!(u);
let pde = Pde::new(equation, u, vec![t, x]);

// Solve
let result = method_of_characteristics(&pde).unwrap();

// Apply IC and verify
let solution = expr!(f(x - c*t));  // Example for transport

}

# Define symbols
u = symbol('u')
t = symbol('t')
x = symbol('x')

# Build PDE
equation = expr(u)
pde = Pde.new(equation, u, [t, x])

# Solve
result = method_of_characteristics(pde)

# Apply IC and verify
solution = expr(f(x - c*t))  # Example for transport

// Define symbols
const u = symbol('u');
const t = symbol('t');
const x = symbol('x');

// Build PDE
const equation = expr(u);
const pde = Pde.new(equation, u, [t, x]);

// Solve
const result = methodOfCharacteristics(pde);

// Apply IC and verify
const solution = expr(f(x - c*t));  // Example for transport

Performance

Time Complexity: N/A (Reference guide)

API Reference

• Rust: mathhook_core::pde::method_of_characteristics
• Python: mathhook.pde.method_of_characteristics
• JavaScript: mathhook.pde.method_of_characteristics

See Also

• advanced.pde.method_of_characteristics

• advanced.pde.classification

• advanced.pde.boundary_conditions