Explicit numerical schemes are popular in multiphysics and multiscale simulations, yet their use in stiff problems often requires time steps to be so small as to render simulations over large time horizons infeasible. Exponential time differencing (ETD) has proved to be an efficient scheme for tackling differential operators with linear stiff and nonlinear non-stiff parts. Such natural separation, however, is absent in many important applications, including multiphase flow and transport in porous media. We introduce a strategy for using ETD in such problems and demonstrate its efficiency in numerical experiments. We also compare the ETD performance to that of an explicit scheme. We conclude that the best outcome is achieved by combining ETD with a fourth-order Runge-Kutta method. Although our methodology is demonstrated on two-dimensional multiphase flow in porous media, it is equally applicable to other applications described by parabolic differential equations of this kind.

显式数值方案在多物理场和多尺度模拟中很流行，但它们在刚性问题中的使用通常需要非常小的时间步长，以至于无法在大的时间范围内进行模拟。指数时间差分 (ETD) 已被证明是处理具有线性刚性和非线性非刚性部件的微分算子的有效方案。然而，这种自然分离在许多重要应用中是不存在的，包括多孔介质中的多相流动和传输。我们介绍了在此类问题中使用 ETD 的策略，并在数值实验中证明了其效率。我们还将 ETD 性能与显式方案的性能进行了比较。我们得出结论，最好的结果是通过将 ETD 与四阶 Runge-Kutta 方法相结合来实现。