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In this study, a numerical solution for degenerate space–time fractional advection–dispersion equations is proposed to simulate atmospheric dispersion in vertically inhomogeneous planetary boundary layers. The fractional derivative exists in a Caputo sense. We establish the maximum principle and a priori estimates for the solutions. Then, we construct a positivity-preserving finite-difference scheme, using monotone discretization in space and L1 approximation on the non-uniform mesh for the time derivative. We use appropriate grading techniques for the time–space mesh in order to overcome the boundary degeneration and weak singularity of the solution at the initial time. The computational results are demonstrated on the Gaussian fractional model as well on the boundary layers defined by height-dependent wind flow and diffusitivity, especially for the Monin–Obukhov model. [ABSTRACT FROM AUTHOR]

The focus of the present work is a one-dimensional system of weakly coupled degenerate semi-linear parabolic equations of optimal portfolio in a regime-switching with power utility function, suggested in Valdez and Vargiolu [Optimal portfolio in a regime-switching model, in Proceedings of the Ascona '11 Seminar on Stochastic Analysis, Random Fields and Applications, R.C. Dalang, M. Dozzi, and F. Russo, eds., 2013, pp. 435–449]. We extend the model, deriving a new linear parabolic system for indifference option valuation. Then, we investigate the properties of the solution of the systems. Further, we develop and analyse an adequate, flux limited finite difference discretization, which preserves the typical features of the differential problem solution. Numerical experiments are presented and discussed. [ABSTRACT FROM AUTHOR]

We present a large class of nonlinear models of European options as parabolic equations with quasi-linear diffusion and fully nonlinear hyperbolic part. The main idea of the operator splitting method (OSM) is to couple known difference schemes for nonlinear hyperbolic equations with other ones for quasi-linear parabolic equations. We use flux limiter techniques, explicit–implicit difference schemes, Richardson extrapolation, etc. Theoretical analysis for illiquid market model is given. The numerical experiments show second-order accuracy for the numerical solution (the price) and Greeks Delta and Gamma, positivity and monotonicity preserving properties of the approximations. [ABSTRACT FROM AUTHOR]

We construct a first order in time and second order in space, positivity preserving numerical method for a generalized Hoggard–Whalley–Wilmott, Leland’s model. We develop the hyperbolic–parabolic operator splitting method, using a kernel based algorithm for the parabolic part and van Leer flux limiter approach for the hyperbolic sub-problem. Properties of the proposed algorithms are discussed. Various numerical examples confirm the efficiency of the proposed method and verify the theoretical statements. [ABSTRACT FROM AUTHOR]

Abstract: In this work we consider Cauchy problem for the so called Gamma equation, which can be derived by transforming the fully nonlinear Black–Scholes equation for option price into a quasi-linear parabolic equation for the second derivative (Greek) of the option price V. We develop an efficient positivity preserving explicit numerical method for solving the model problem concerning different volatility terms. We prove that the obtained semi-discretization is positive and the corresponding full approximation also preserves this property, if the time step is restricted. The stability of the difference scheme in norm is shown and the existence of interface curves is investigated numerically. Results of numerical simulations are given and discussed. [Copyright &y& Elsevier]

In this paper, we consider models of cancer migration and invasion, which consist of two nonlinear parabolic equations (one of the convection–diffusion reaction type and the other of the diffusion–reaction type) and an additional nonlinear ordinary differential equation. The unknowns represent concentrations or densities that cannot be negative. Widely used approximations, such as difference schemes, can produce negative solutions because of truncation errors and can become unstable. We propose a new difference scheme that guarantees the positivity of the numerical solution for arbitrary mesh step sizes. It has explicit and fast performance even for nonlinear reaction terms that consist of sums of positive and negative functions. The numerical examples illustrate the simplicity and efficiency of the method. A numerical simulation of a model of cancer migration is also discussed. [ABSTRACT FROM AUTHOR]

We consider one-dimensional systems of weakly coupled degenerate semi-linear parabolic equations for buyer's and writer's pricing of European options under regime-switching with exponential utility function. First, we prove a comparison principle and then we establish a maximum principle for the indifference pricing differential problems. The differential problems are solved numerically by fitted finite volume method with second-order of accuracy. We prove the discrete maximum principle and convergence of the numerical solutions in maximum norm. Numerical results, illustrating the theoretical statements are presented and discussed. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel exponential difference scheme for solving non-linear problems arising in variably saturated flow with discontinuous absolute permeability. First, we derive the discretization for a linear convection-diffusion-reaction problem with discontinuous coefficients. Positivity preserving property, stability and convergence of the scheme are studied. Then, the method is implemented to the transport equation and Richards' equation. Various numerical experiments on graded and uniform meshes are presented and discussed. [ABSTRACT FROM AUTHOR]