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We consider the numerical approximation of a general second ordersemi--linear parabolic partial differential equation. Equations of this typearise in many contexts, such as transport in porous media which is fundamentalin many geo-engineering applications, including oil and gas recovery fromsubsurface. Using the finite volume with two-point flux approximation onregular mesh combined with exponential time differencing of order one (ETD1)for temporal discretization, we derive the $L^{2}$ estimate under theassumption that the non linear term is locally Lipschitz. Numerical simulationsto sustain the theoretical results are provided.

An exponential Integrator for finite volume discretization of nonlinear parabolic differential equation