What are the tradeoffs between implicit and explicit finite difference schemes?

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AcadiFi · Accepted Answer

Explicit FD computes each new node directly from known adjacent nodes at the previous time step - simple, fast per step, but conditionally stable. The stability condition requires dt <= dS^2 / (sigma^2 * S_max^2), which is brutal: halving dS requires quartering dt, blowing up compute time. If violated, the solution explodes with oscillations. Explicit schemes look nearly identical to trinomial trees mathematically. Implicit FD solves a tridiagonal system at each time step - each new node depends on three unknown neighbors at the new time level. It's unconditionally stable (any dt works), making it robust, but requires solving a linear system (O(M) per step via Thomas algorithm). First-order accurate in time. For a valuation of Yarrow Minerals convertible bond with complex features, explicit FD might need 50,000 time steps to remain stable, while implicit needs only 300-500. Crank-Nicolson gets the best of both: unconditional stability plus second-order time accuracy. ```mermaid graph TD A[Explicit] --> B[Fast per step
unstable] C[Implicit] --> D[Solve system
stable] E[CN blend] --> F[Best accuracy] ```

What are the tradeoffs between implicit and explicit finite difference schemes?

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