Positivity for Convective Semi-discretizations

Abstract

We propose a technique for investigating stability properties like positivity and forward invariance of an interval for method-of-lines discretizations, and apply the technique to study positivity preservation for a class of TVD semi-discretizations of 1D scalar hyperbolic conservation laws. This technique is a generalization of the approach suggested in Khalsaraei (J Comput Appl Math 235(1): 137–143, 2010). We give more relaxed conditions on the time-step for positivity preservation for slope-limited semi-discretizations integrated in time with explicit Runge–Kutta methods. We show that the step-size restrictions derived are sharp in a certain sense, and that many higher-order explicit Runge–Kutta methods, including the classical 4th-order method and all non-confluent methods with a negative Butcher coefficient, cannot generally maintain positivity for these semi-discretizations under any positive step size. We also apply the proposed technique to centered finite difference discretizations of scalar hyperbolic and parabolic problems.

Appendix: Some Mathematica Code

1.1 Generating the Multivariable Polynomials

The first cell below contains the definition of a Mathematica function ERKpolynomials for generating the multivariable polynomials in (14). The two arguments A and b correspond to the Butcher tableau of the ERK method, and the output is a list of the $m+1$ polynomials $P_0$, ..., $P_m$ in the variables $\xi _\ell ^j$. Note that the superscripts in $\xi _\ell ^j$ are not exponents; the symbols $\xi _\ell ^j$ with different sub- or superscripts denote different variables.

The second cell illustrates how to obtain the $4+1=5$ polynomials in $\frac{4\cdot (4+1)}{2}=10$ variables corresponding to the classical ERK(4,4) method.

1.2 Non-negativity of Polynomials at the Vertices of a Hypercube

Here we provide a simple Mathematica code to test the non-negativity of a multivariable polynomial by evaluating it at each vertex of a hypercube. In this particular example, the polynomial $P_{-1}(x,y,z,u)$ from Sect. 6.2 is evaluated at the $2^4$ vertices of the hypercube $[0,\varepsilon ]^4$ with some $\varepsilon >0$, and the resulting system of 16 inequalities $P_{-1}\ge 0$ is solved.