Max Cubillos

Max Cubillos

The multiplicative (or geometric) calculus is a non-Newtonian calculus derived from an arithmetic in which the operations of addition/subtraction/multiplication are replaced by multiplication/division/exponentiation. A major difference between the multiplicative calculus and the classical additive calculus, and one that has important consequences in the simulation of wave propagation problems, is that in geometric calculus the role of polynomials is played by exponentials of a polynomial argument. For example, whereas a polynomial of degree one has constant (classical) derivative, it is the exponential function that has constant derivative in the multiplicative calculus. As we will show, this implies that even low-order finite quotient approximations|the analogues of finite differences in the multiplicative calculus|produce exact multiplicative derivatives of exponential functions. We exploit this fact to show that some partial differential equations (PDE) can be solved far more efficiently using techniques based on the multiplicative calculus. For wave propagation models in particular, we will show that it is possible to circumvent the minimum-points-per-wavelength sampling constraints of classical methods. In this first part we develop the theoretical framework for studying multiplicative partial differential equations and their connections with classical models.

Oscar Bruno, Max Cubillos

The companion paper "Higher-order in time quasi-unconditionally stable ADI solvers for the compressible Navier-Stokes equations in 2D and 3D curvilinear domains", which is referred to as Part I in what follows, introduces ADI (Alternating Direction Implicit) solvers of higher orders of temporal accuracy (orders $s = 2$ to $6$) for the compressible Navier-Stokes equations in two- and three-dimensional space. The proposed methodology employs the backward differentiation formulae (BDF) together with a quasilinear-like formulation, high-order extrapolation for nonlinear components, and the Douglas-Gunn splitting. A variety of numerical results presented in Part I demonstrate in practice the theoretical convergence rates enjoyed by these algorithms, as well as their excellent accuracy and stability properties for a wide range of Reynolds numbers. In particular, the proposed schemes enjoy a certain property of "quasi-unconditional stability": for small enough (problem-dependent) fixed values of the time-step $Δt$, these algorithms are stable for arbitrarily fine spatial discretizations. The present contribution presents a mathematical basis for the performance of these algorithms. Short of providing stability theorems for the full BDF-ADI Navier-Stokes solvers, this paper puts forth proofs of unconditional stability and quasi-unconditional stability for BDF-ADI schemes as well as some related un-split BDF schemes, for a variety of related linear model problems in one, two and three spatial dimensions, and for schemes of orders $2\leq s\leq 6$ of temporal accuracy. Additionally, a set of numerical tests presented in this paper for the compressible Navier-Stokes equation indicate that quasi-unconditional stability carries over to the fully non-linear context.

Oscar Bruno, Max Cubillos

This paper introduces alternating-direction implicit (ADI) solvers of higher order of time-accuracy (orders two to six) for the compressible Navier-Stokes equations in two- and three-dimensional curvilinear domains. The higher-order accuracy in time results from 1) An application of the backward differentiation formulae time-stepping algorithm (BDF) in conjunction with 2) A BDF-like extrapolation technique for certain components of the nonlinear terms (which makes use of nonlinear solves unnecessary), as well as 3) A novel application of the Douglas-Gunn splitting (which greatly facilitates handling of boundary conditions while preserving higher-order accuracy in time). As suggested by our theoretical analysis of the algorithms for a variety of special cases, an extensive set of numerical experiments clearly indicate that all of the BDF-based ADI algorithms proposed in this paper are "quasi-unconditionally stable" in the following sense: each algorithm is stable for all couples $(h,Δt)$ of spatial and temporal mesh sizes in a problem-dependent rectangular neighborhood of the form $(0,M_h)\times (0,M_t)$. In other words, for each fixed value of $Δt$ below a certain threshold, the Navier-Stokes solvers presented in this paper are stable for arbitrarily small spatial mesh-sizes. The second order formulation has further been rigorously shown to be unconditionally stable for linear hyperbolic and parabolic equations in two-dimensional space. Although implicit ADI solvers for the Navier-Stokes equations with nominal second-order of temporal accuracy have been proposed in the past, the algorithms presented in this paper are the first ADI-based Navier-Stokes solvers for which second order or better accuracy has been verified in practice under non-trivial (non-periodic) boundary conditions.