Lattice approximation for stochastic reaction diffusion equations with one-sided Lipschitz condition

Abstract

We consider strong convergence of the finite differences approximation in space for stochastic reaction diffusion equations in one dimension with multiplicative noise under a one-sided Lipschitz condition only. The equation may be additionally coupled with a noisy control variable with global Lipschitz condition but no diffusion. We derive convergence with an implicit rate depending on the regularity of the exact solution. This can be made explicit if the variational solution has more than its canonical spatial regularity. As an application, spatially extended FitzHugh-Nagumo systems with noise are considered.

Analysis and approximation of stochastic nerve axon equations

“…Section 3 is then devoted to the existence and uniqueness Theorem 3.1, while Section 4 introduces the approximation scheme as well as states Theorems 4.1 and 4.3 on convergence and explicit rates. We finish with two examples, on the one hand the Hodgkin-Huxley system mentioned before and also the FitzHugh-Nagumo equations studied in [27]. In particular, we are able to generalize and improve the results obtained there.…”

“…We deduce explicit error estimates, which a priori do not yield a strong convergence rate but only pathwise convergence with smaller rate 1 /2−. In special cases, e. g. when the drift satisfies a one-sided Lipschitz condition as in [27], one can improve the result to obtain a strong convergence rate of 1 /n.…”

“…Concerning the numerical approximation we consider the well-known finite difference method for the spatial variable only. This method has been studied intensively, e. g. by Gyöngy [10], Shardlow [28], Pettersson & Signahl [25] and more recently by the authors [27] applied to the simpler FitzHugh-Nagumo equations. Although the method is heavily used in applied sciences, up to the best of our knowledge none of the existing literature covers a convergence result with explicit rates for such non-globally Lipschitz and non-monotone coefficients.…”

Reliability of signal transmission in stochastic nerve axon equations

“…with Neumann boundary conditions as in (2). The numerical method chosen for the integration of such a SPDE is a finite difference approximation in both space and time, see Sauer & Stannat (2015, 2014 for details. For the space variable x we use an equidistant grid (x i ) of size ∆x = L /N and replace the second derivative by its two-sided difference quotient.…”

Optimal Strong Rates of Convergence for a Space-Time Discretization of the Stochastic Allen–Cahn Equation with Multiplicative Noise

“…Next to it, we use again (1.3) for the drift, in combination with wellknown approximation results for a finite element discretization to show that the error part due to spatial discretization is of order O( √ k + h) where k > 0 is the time discretization parameter and h > 0 is the space discretization parameter (see Theorem 5.2). In this context, we mention the numerical analysis in [11] for an extended model of (1.2), where the uniform bounds for the exponential moments next to arbitrary moments in stronger norms are obtained for the solution of a semi-discretization in space in the case d = 1 (see [11,Prop.s 4.2,4.3]); those bounds, together with a monotonicity argument are then used to properly address the nonlinear effects in the error analysis and arrive at the (lower) strong rate 1 2 for the p-th mean convergence of the numerical solution.…”