An exponential Integrator for finite volume discretization of nonlinear parabolic differential equation
Provides theoretical convergence guarantees for a numerical method applied to nonlinear parabolic PDEs, relevant to computational scientists in geo-engineering.
The authors develop an exponential time differencing method (ETD1) for finite volume discretization of nonlinear parabolic PDEs, proving L² error estimates under local Lipschitz conditions and validating with numerical simulations.
We consider the numerical approximation of a general second order semi--linear parabolic partial differential equation. Equations of this type arise in many contexts, such as transport in porous media which is fundamental in many geo-engineering applications, including oil and gas recovery from subsurface. Using the finite volume with two-point flux approximation on regular mesh combined with exponential time differencing of order one (ETD1) for temporal discretization, we derive the $L^{2}$ estimate under the assumption that the non linear term is locally Lipschitz. Numerical simulations to sustain the theoretical results are provided.