y'=f(t,y). Euler O(h), RK4 O(h⁴). Consistency, convergence, stability (A-stability). Stiff equations.
graph TD
D1["Def: One-step method\nyₙ₊₁ = Φ(tₙ,yₙ,h)"]
D2["Def: Order of accuracy\nlocal truncation O(hᵖ⁺¹)"]
D3["Def: A-stable\nReλ<0 ⇒ |R(z)|≤1"]
T1["Thm: Euler\norder 1"]
T2["Thm: RK4\norder 4, 4 stages"]
T3["Thm: Convergence\nconsistency + stability"]
T4["Thm: Dahlquist\nA-stable impl ⇒ order ≤2"]
T5["Thm: B-stability\ncontractive for dissipative"]
L1["Lemma: Butcher tableau"]
D1 --> D2
D1 --> T1
D1 --> T2
D2 --> T3
D3 --> T4
T2 --> L1
T3 --> T5
classDef definition fill:#b197fc,color:#fff
classDef theorem fill:#51cf66,color:#fff
classDef lemma fill:#74c0fc,color:#fff
class D1,D2,D3 definition
class T1,T2,T3,T4,T5 theorem
class L1 lemma
Process Statistics
- Nodes: 15
- Edges: 12
- Definitions: 3
- Theorems: 5
- Lemmas: 1
Frontier: Structure-preserving (symplectic, energy), geometric integration. math.NA