Nonstandard finite difference schemes for the Black-Scholes partial differential equation preserving the positivity property are proposed. Computationally simple schemes are derived by using a nonlocal approximation in the reaction term of the Black-Scholes equation. Unlike the standard methods, the solutions of new proposed schemes are positive and free of the spurious oscillations.

Patankar-type schemes are linearly implicit time integration methods designed to be unconditionally positivity-preserving. However, there only little results on their stability or robustness. We suggest two approaches analyze the performance and robustness of these methods. In particular, we demonstrate problematic behaviors that, even very simple linear problems, can lead undesired oscillation...

We further investigate the high order positivity-preserving discontinuous Galerkin (DG) methods for linear hyperbolic and radiative transfer equations developed in [14]. The DG methods in [14] can maintain positivity and high order accuracy, but they rely both on the scaling limiter in [15] and a rotational limiter, the latter may alter cell averages of the unmodulated DG scheme, thereby affect...

Lubrication equations are fourth order degenerate diffusion equations of the form ht + ∇ · (f(h)∇∆h) = 0, describing thin films or liquid layers driven by surface tension. Recent studies of singularities in which h → 0 at a point, describing rupture of the fluid layer, show that such equations exhibit complex dynamics which can be difficult to simulate accurately. In particular, one must ensure...

Functions preserving Loewner positivity when applied entrywise to positive semidefinite matrices have been widely studied in the literature. Following the work of Schoenberg [Duke Math. J. 9], Rudin [Duke Math. J. 26], and others, it is well-known that functions preserving positivity for matrices of all dimensions are absolutely monotonic (i.e., analytic with nonnegative Taylor coefficients). I...