Numerical solution of degenerate stochastic Kawarada equations via a semi-discretized approach

Abstract

The numerical solution of a highly nonlinear two-dimensional degenerate stochastic Kawarada equation is investigated. A semi-discretized approximation in space is comprised on arbitrary nonuniform grids. Exponential splitting strategies are then applied to advance solutions of the semi-discretized scheme over adaptive grids in time. It is shown that key quenching solution features including the positivity and monotonicity are well preserved under modest restrictions. The numerical stability of the underlying splitting method is also maintained without any additional restriction. Computational experiments are provided to not only illustrate our results, but also provide further insights into the global nonlinear convergence of the numerical solution.

An Exploration of a Balanced Up-downwind Scheme for Solving Heston Volatility Model Equations on Variable Grids

“…Our ongoing research has been including effective schemes on variable spacial and temporal meshes for different financial products and simulations. We have also been considering effective adaptation strategies such as those investigated in [16,18].…”

“…Some key parameters used are shown in Table 1. Further, a Crank-Nicolson type temporal integrator will be utilized for advancing our semi-discretized system (2.9), (2.12), with ∆τ as the temporal step [18]. It has been known that λ = ∆τ/c 2 , where c = min {h, k} , play an effective role of the Courant number [14,16].…”

“…The principal is commonly utilized in adaptive computations and serves as an initial exploration for more sophisticated adaptations [3,20]. The calculation of the mesh coordinates in our experiments is conducted based on a forward Euler formula for arc-lengths [18]. For instance, in the S-direction we have…”

The quenching of solutions to time–space fractional Kawarada problems

“…Classical studies regarding the integer order Kawarada equations have often considered the solution positivity and monotonicy [15,17,16,8]. These properties are critical in many situations modeled by these equations, such as solid-fuel combustion [3].…”

Convergence of an Operator Splitting Scheme for Abstract Stochastic Evolution Equations

“…In particular, operator splitting methods have been shown to be effective and efficient numerical methods, as they may often be constructed to preserve stability while being explicit with desirable convergence rates [9,10,19,20,23,24]. While splitting methods have primarily been studied in the deterministic setting, there have been several recent studies regarding their efficacy in application to stochastic problems [2,17,18,21]. In particular, it has been shown that the splitting of deterministic and stochastic counterparts of differential equations can prove effective by increasing convergence rates without the inclusion of derivative terms [2,5,17].…”