Overview
The implicit Euler method is a first-order implicit integrator well-suited for stiff ordinary differential equations. It offers improved stability over explicit schemes, making it a preferred choice in many engineering simulations.
Mathematical Formulation
Given dy/dt = f(t,y), implicit Euler advances the solution as y_{n+1} = y_n + h f(t_{n+1}, y_{n+1}). This requires solving a generally nonlinear equation at each step.
Newton Iteration
We solve the implicit equation using Newton-Raphson: solve g(y)=y - y_n - h f(t_{n+1}, y)=0. The Jacobian is approximated numerically for robustness.
Jacobian Approximation
Analytic Jacobians improve convergence but may not be available. A centered finite difference approximates df/dy, trading a small cost for generality and reliability across problems.
Step Control and Stability
Although implicit Euler is A-stable, step size affects accuracy. For stiff problems, larger steps remain stable but may lose accuracy. Use smaller steps or adaptivity when accuracy is the priority.
Practical Implementation
In practice, use an initial guess from explicit Euler, then iterate Newton updates until convergence. Safeguards such as max iterations and tolerances prevent infinite loops.
Performance Considerations
Newton iterations and Jacobian approximations increase per-step cost. However, for stiff systems the ability to use larger steps typically offsets this overhead.
Use Cases
Implicit Euler excels in chemical kinetics, circuit simulation, damped mechanical systems, and any stiff ODE where explicit integrators require prohibitively small time steps.