Nonmonotone chemotactic invasion: High-resolution simulations, phase plane analysis and new benchmark problems

Verifying and validating the performance of a computational simulation is a challenging problem of interest in all applied sciences [12,13]. Measuring the performance of a numerical algorithm against standard benchmark test cases is often the first step in assessing algorithm performance. These test cases are used repeatedly over long periods of time as algorithms are devised and improved. In addition, new benchmark test cases are continually developed [9]. Effective benchmarks are characterized by having either a complete or an approximate analytical solution against which the numerical results can be quantitatively compared [1,5]. It is also possible, although less common, to use carefully collected laboratory data to benchmark a numerical code [18,22]. To identify an effective benchmark test case, a balance must be found between two opposing requirements: the benchmark ought to be sufficiently complex to rigorously challenge a numerical algorithm, and it should also be amenable to analysis. Certain problems have become synonymous with algorithm development in various disciplines. Particular problems from fluid mechanics, for example, are associated with long-celebrated benchmarks. Algorithms designed to solve steady incompressible Navier–Stokes flows are often tested with lid-driven cavity flow problems and Burggraf’s solution [1,3]. Algorithms designed to solve convectively-driven flow in porous media are usually benchmarked with Henry’s solution for salt water intrusion [5,19]. Many other examples of popular benchmark test cases can be found in the literature. Constructing numerical algorithms to accurately approximate the solution of hyperbolic conservation laws (HCL) is of wide interest and challenging. Typically HCL algorithms are tested with a suite of benchmark problems starting with single species linear advection [2,3,6,10]. Linear advection has the advantage of being conceptually simple, analytically tractable and clearly identifies susceptibility to numerical diffusion and

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