timemarching

Timemarching is a numerical technique for solving time-dependent problems by advancing the solution in discrete time steps. It is commonly used for initial-value problems governed by ordinary differential equations or partial differential equations, where the state of the system at time t is propagated to time t+Δt through an evolution operator derived from the governing equations.

In explicit timemarching, the next state is computed directly from known quantities at the current step, requiring no solution of algebraic systems. In implicit timemarching, the next state appears on both sides of the update and typically requires solving a nonlinear or linear system. Semi-implicit and operator-splitting methods blend features of both. Time discretizations range from first-order forward Euler and backward Euler to higher-order schemes such as Runge–Kutta methods and the Crank–Nicolson method.

Stability and accuracy depend on the time step and the discretization. Explicit schemes can be conditionally stable, subject to the Courant–Friedrichs–Lewy condition, especially for hyperbolic problems. Implicit schemes handle stiffness better but require more computation per step. Adaptive timestepping adjusts Δt based on error estimates to balance efficiency and accuracy.

Timemarching is ubiquitous in computational fluid dynamics, weather and climate modeling, acoustics and electromagnetics simulations, structural dynamics, and the numerical solution of the time-dependent Schrödinger equation. It is often contrasted with frequency-domain or steady-state solvers that seek solutions without time evolution.

As a broad class, timemarching encompasses various schemes and strategies for marching the solution forward in time, with choices driven by stability, stiffness, accuracy, and computational resources.