Positivity Preserving Research Articles

A limiter‐based well‐balanced discontinuous Galerkin method for shallow‐water flows with wetting and drying: Triangular grids

SummaryA novel wetting and drying treatment for second‐order Runge‐Kutta discontinuous Galerkin methods solving the nonlinear shallow‐water equations is proposed. It is developed for general conforming two‐dimensional triangular meshes and utilizes a slope limiting strategy to accurately model inundation. The method features a nondestructive limiter, which concurrently meets the requirements for linear stability and wetting and drying. It further combines existing approaches for positivity preservation and well balancing with an innovative velocity‐based limiting of the momentum. This limiting controls spurious velocities in the vicinity of the wet/dry interface. It leads to a computationally stable and robust scheme, even on unstructured grids, and allows for large time steps in combination with explicit time integrators. The scheme comprises only one free parameter, to which it is not sensitive in terms of stability. A number of numerical test cases, ranging from analytical tests to near‐realistic laboratory benchmarks, demonstrate the performance of the method for inundation applications. In particular, superlinear convergence, mass conservation, well balancedness, and stability are verified.

A Positivity Preserving and Free Energy Dissipative Difference Scheme for the Poisson–Nernst–Planck System

In this paper, based on the reformulation of the Nernst–Planck equations, we construct an unconditionally stable semi-implicit linearized difference scheme for the time dependent Poisson–Nernst–Planck system. The method preserves several important physical laws at full discrete level without any constraints on the time step size, including: mass conservation, ion concentration positivity, free energy dissipation. The proposed method is a linearized method, which can be implemented efficiently. Furthermore, the method can be extended to the case of multi-ions without any difficulty. The paper ends with supporting numerical results which confirm well with theoretical findings.

On the ground state for quantum graphs

Ground-state eigenfunctions of Schrödinger operators can often be chosen positive. We analyse to which extent this is true for quantum graphs—differential operators on metric graphs. It is shown that the theorem holds in the case of generalised delta couplings at the vertices—a new class of vertex conditions introduced in the paper. It is shown that this class of vertex conditions is optimal. Relations to positivity preserving and positivity improving semigroups are clarified.

Comment on “A positivity preserving and well-balanced DG scheme using finite volume subcells in almost dry regions”

This comment concerns a mistake in the proving of positivity-preserving property for the well-balanced discontinuous Galerkin methods presented by Meister and Ortleb [Meister A, Ortleb S, A positivity preserving and well-balanced DG scheme using finite volume subcells in almost dry regions, Applied Mathematics and Computation 272 (2016) 259–273]. Furthermore, we provide a correct proof which gives the same result.

Provably positive high-order schemes for ideal magnetohydrodynamics: analysis on general meshes

This paper proposes and analyzes arbitrarily high-order discontinuous Galerkin (DG) and finite volume methods which provably preserve the positivity of density and pressure for the ideal MHD on general meshes. Unified auxiliary theories are built for rigorously analyzing the positivity-preserving (PP) property of MHD schemes with a HLL type flux on polytopal meshes in any space dimension. The main challenges overcome here include establishing relation between the PP property and discrete divergence of magnetic field on general meshes, and estimating proper wave speeds in the HLL flux to ensure the PP property. In 1D case, we prove that the standard DG and finite volume methods with the proposed HLL flux are PP, under condition accessible by a PP limiter. For multidimensional conservative MHD system, standard DG methods with a PP limiter are not PP in general, due to the effect of unavoidable divergence-error. We construct provably PP high-order DG and finite volume schemes by proper discretization of symmetrizable MHD system, with two divergence-controlling techniques: locally divergence-free elements and a penalty term. The former leads to zero divergence within each cell, while the latter controls the divergence error across cell interfaces. Our analysis reveals that a coupling of them is important for positivity preservation, as they exactly contribute the discrete divergence-terms absent in standard DG schemes but crucial for ensuring the PP property. Numerical tests confirm the PP property and the effectiveness of proposed PP schemes. Unlike conservative MHD system, the exact smooth solutions of symmetrizable MHD system are proved to retain the positivity even if the divergence-free condition is not satisfied. Our analysis and findings further the understanding, at both discrete and continuous levels, of the relation between the PP property and the divergence-free constraint.

Dirichlet Parabolic Problems Involving Schrödinger Type Operators with Unbounded Diffusion and Singular Potential Terms in Unbounded Domains

We study the well-posedness of autonomous parabolic Dirichlet problems involving Schrodinger type operators of the form $$\begin{aligned} H_{\alpha ,a,b,c}=(1+|x|^\alpha )\Delta +a|x|^\alpha +b|x|^{\alpha -2}+c|x|^{-2}, \end{aligned}$$ with $$\alpha \ge 0$$ , $$a<0$$ and $$b,c\in \mathbb {R}$$ , in regular unbounded domains $$\Omega \subset \mathbb {R}^N$$ containing 0. Under suitable assumptions on $$\alpha $$ , b and c, the solution is governed by a contractive and positivity preserving strongly continuous (analytic) semigroup on the weighted space $$L^p(\Omega , d\mu (x))$$ , $$1<p<\infty $$ , where $$d\mu (x)=(1+|x|^\alpha )^{-1}dx$$ . The proofs are based on some $$L^p$$ -weighted Hardy’s inequality and perturbation techniques.

High-Order Lax-Friedrichs WENO Fast Sweeping Methods for the SN Neutron Transport Equation

Fast sweeping methods are efficient iterative techniques originally developed to solve the steady-state Hamilton-Jacobi equations and later used for the hyperbolic conservation laws. For these boundary value problems, their solution information propagates along characteristics starting from the boundary. These fast sweeping methods take advantage of this property and achieve very fast convergence based on a Gauss-Seidel–type iteration approach and alternating-direction sweeping strategy. In this paper, we solve the SN neutron transport equation using the high-order Lax-Friedrichs Weighted Essentially Non-Oscillatory (LF-WENO) fast sweeping methods. Our numerical tests in one and two dimensions demonstrate that the proposed new sweeping methods can achieve better accuracy and positivity preserving than the diamond difference method for the SN solution.

A conservative, free energy dissipating, and positivity preserving finite difference scheme for multi-dimensional nonlocal Fokker–Planck equation

In this work, we design and analyze a conservative, positivity preserving, and free energy dissipating finite difference method for the multi-dimensional nonlocal Fokker–Planck (FP) equation. Based on a non-logarithmic Landau transformation, a central-differencing spatial discretization using harmonic-mean approximations is developed. Both forward and backward Euler discretizations in time are employed to derive an explicit scheme and a linearized semi-implicit scheme, respectively. Three desired properties that are possessed by analytical solutions: i) mass conservation, ii) free-energy dissipation, and iii) positivity, are proved to be maintained at discrete level. Remarkably, numerical analysis demonstrates that the semi-implicit time discretization ensures the property of positivity preserving unconditionally. Due to the advantages brought by the harmonic-mean approximations, an estimate on the upper bound of the condition number of the resulting coefficient matrix is further established for the semi-implicit scheme. Extensive numerical tests are performed to validate aforementioned properties numerically.

Numerical simulations for the chemotaxis models on surfaces via a novel characteristic finite element method

In this paper, the chemotaxis partial differential equations models which describe the movement by one community in reaction to one chemical or biological signal are given and solved numerically on surfaces. The models and the numerical method concerned are in terms of a generic form of chemotaxis models. We develop a new semi-implicit finite element scheme based on the gradient and Laplacian recoveries to linearize the equations. Meanwhile, we modify the proposed semi-implicit scheme as a characteristic form and present a novel strategy for the discretization of characteristic derivative on surface. The lumped mass modification is employed for positivity preservation. And the related analysis results are provided. We investigate the accuracy and convergence of the proposed method by numerical tests. And the simulations of blowing-up solution, pattern formulations and aggregations of bacteria demonstrate the applicability of the proposed methods.

An entropy stable discontinuous Galerkin method for the shallow water equations on curvilinear meshes with wet/dry fronts accelerated by GPUs

We extend the entropy stable high order nodal discontinuous Galerkin spectral element approximation for the non-linear two dimensional shallow water equations presented by Wintermeyer et al. (2017) [46] with a shock capturing technique and a positivity preservation capability to handle dry areas. The scheme preserves the entropy inequality, is well-balanced and works on unstructured, possibly curved, quadrilateral meshes. For the shock capturing, we introduce an artificial viscosity to the equations and prove that the numerical scheme remains entropy stable. We add a positivity preserving limiter to guarantee non-negative water heights as long as the mean water height is non-negative. We prove that non-negative mean water heights are guaranteed under a certain additional time step restriction for the entropy stable numerical interface flux. We implement the method on GPU architectures using the abstract language OCCA, a unified approach to multi-threading languages. We show that the entropy stable scheme is well suited to GPUs as the necessary extra calculations do not negatively impact the runtime up to reasonably high polynomial degrees (around N=7). We provide numerical examples that challenge the shock capturing and positivity properties of our scheme to verify our theoretical findings.

On the loss of maximum principles for higher-order fractional Laplacians

We study existence, regularity, and qualitative properties of solutions to linear problems involving higher-order fractional Laplacians $(-\Delta)^s$ for any $s>1$. Using the nonlocal properties of these operators, we provide an explicit counterexample to general maximum principles for $s\in(n,n+1)$ with $n\in\mathbb N$ odd; moreover, using a representation formula for solutions, we derive regularity and positivity preserving properties whenever the domain is the whole space or a ball. In the case of the whole space we analyze the Riesz kernel, which provides a fundamental solution, while in the case of the ball we show the validity of Boggio's representation formula for all integer and fractional powers of the Laplacian $s>0$. Our proofs rely on characterizations of $s$-harmonic functions using higher-order Martin kernels, on a decomposition of Boggio's formula, and on elliptic regularity theory.

Positivity Preserving Gradient Approximation with Linear Finite Elements

Abstract Preserving positivity precludes that linear operators onto continuous piecewise affine functions provide near best approximations of gradients. Linear interpolation thus does not capture the approximation properties of positive continuous piecewise affine functions. To remedy, we assign nodal values in a nonlinear fashion such that their global best error is equivalent to a suitable sum of local best errors with positive affine functions. As one of the applications of this equivalence, we consider the linear finite element solution to the elliptic obstacle problem and derive that its error is bounded in terms of these local best errors.

Green function and Martin kernel for higher-order fractional Laplacians in balls

We give the explicit formulas for the Green function and the Martin kernel for all integer and fractional powers of the Laplacian s>1 in balls. As consequences, we deduce interior and boundary regularity estimates for solutions to linear problems and positivity preserving properties. Our proofs rely on a characterization of suitable s-harmonic functions and on a differential recurrence equation.

Anti-symmetric plasma moment equations with conservative discrete counterparts

We derive a set of fluid moment equations with inherent consistency and numerical stability, conceived by exploiting the anti-symmetric nature of the plasma flow operator (∇·v+v·∇). The obtained equations can be interpreted as an alternative to the traditional Eulerian and Lagrangian representations—one in which plasma flows generate infinitesimal rotations of generalized fluid moments n, nv, and p. The continuous model has a discrete analog with exact mass, momentum, and energy conservation, which is achieved by construction through vanishing integrals of the anti-symmetric flow terms. Positivity preservation is obtained through the use of the generalized moment quantities. The conservation properties of the approach are illustrated in simulations of seeded blob propagation, where we verify numerical conservation to machine accuracy.

Reprint of: Boundary conditions for fractional diffusion

This paper derives physically meaningful boundary conditions for fractional diffusion equations, using a mass balance approach. Numerical solutions are presented, and theoretical properties are reviewed, including well-posedness and steady state solutions. Absorbing and reflecting boundary conditions are considered, and illustrated through several examples. Reflecting boundary conditions involve fractional derivatives. The Caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion, since the resulting boundary value problem is not positivity preserving.

A positivity preserving and conservative variational scheme for phase-field modeling of two-phase flows

We present a positivity preserving variational scheme for the phase-field modeling of incompressible two-phase flows with high density ratio. The variational finite element technique relies on the Allen–Cahn phase-field equation for capturing the phase interface on a fixed Eulerian mesh with mass conservative and energy-stable discretization. The mass conservation is achieved by enforcing a Lagrange multiplier which has both temporal and spatial dependence on the underlying solution of the phase-field equation. To make the scheme energy-stable in a variational sense, we discretize the spatial part of the Lagrange multiplier in the phase-field equation by the mid-point approximation. The proposed variational technique is designed to reduce the spurious and unphysical oscillations in the solution while maintaining the second-order accuracy of both spatial and temporal discretizations. We integrate the Allen–Cahn phase-field equation with the incompressible Navier–Stokes equations for modeling a broad range of two-phase flow and fluid-fluid interface problems. The coupling of the implicit discretizations corresponding to the phase-field and the incompressible flow equations is achieved via nonlinear partitioned iterative procedure. Comparison of results between the standard linear stabilized finite element method and the present variational formulation shows a remarkable reduction of oscillations in the solution while retaining the boundedness of the phase-indicator field. We perform a standalone test to verify the accuracy and stability of the Allen–Cahn two-phase solver. We examine the convergence and accuracy properties of the coupled phase-field solver through the standard benchmarks of the Laplace–Young law and a sloshing tank problem. Two- and three-dimensional dam break problems are simulated to assess the capability of the phase-field solver for complex air-water interfaces involving topological changes on unstructured meshes. Finally, we demonstrate the phase-field solver for a practical offshore engineering application of wave-structure interaction.

Positivity-preserving dual time stepping schemes for gas dynamics

A new approach at discretizing the temporal derivative of the Euler equations is here presented which can be used with dual time stepping. The temporal discretization stencil is derived along the lines of the Cauchy–Kowalevski procedure resulting in cross differences in spacetime but with some novel modifications which ensure the positivity of the discretization coefficients. It is then shown that the so-obtained spacetime cross differences result in changes to the wave speeds and can thus be incorporated within Roe or Steger–Warming schemes (with and without reconstruction–evolution) simply by altering the eigenvalues. The proposed approach is advantaged over alternatives in that it is positivity-preserving for the Euler equations. Further, it yields monotone solutions near discontinuities while exhibiting a truncation error in smooth regions less than the one of the second- or third-order accurate backward-difference-formula (BDF) for either small or large time steps. The high resolution and positivity preservation of the proposed discretization stencils are independent of the convergence acceleration technique which can be set to multigrid, preconditioning, Jacobian-free Newton–Krylov, block-implicit, etc. Thus, the current paper also offers the first implicit integration of the time-accurate Euler equations that is positivity-preserving in the strict sense (that is, the density and temperature are guaranteed to remain positive). This is in contrast to all previous positivity-preserving implicit methods which only guaranteed the positivity of the density, not of the temperature or pressure. Several stringent reacting and inert test cases confirm the positivity-preserving property of the proposed method as well as its higher resolution and higher computational efficiency over other second-order and third-order implicit temporal discretization strategies.

On positivity preservation in nonlinear finite volume method for multi-term fractional subdiffusion equation on polygonal meshes

The positivity preserving is one of the key requirements to discrete schemes for subdiffusion equation. The main goal of our research is to explore a spatial second-order nonlinear finite volume (FV) method solving the multi-term time-fractional subdiffusion equation with this property maintained. Compared to the already published results, our findings have two important special features. First, we prove positivity preservation property of the equation. Second, we construct a nonlinear FV method for fractional subdiffusion equation on star-shaped polygonal meshes and prove that it preserves positivity of analytical solutions for strongly anisotropic and heterogeneous full tensor coefficients. Numerical experiments are presented to verify our theoretical findings for both smooth and non-smooth highly anisotropic solutions. Moreover, numerical results show that our scheme has approximate second-order accuracy for the solution and first-order accuracy for the flux on various distorted meshes.

Boundary conditions for fractional diffusion

This paper derives physically meaningful boundary conditions for fractional diffusion equations, using a mass balance approach. Numerical solutions are presented, and theoretical properties are reviewed, including well-posedness and steady state solutions. Absorbing and reflecting boundary conditions are considered, and illustrated through several examples. Reflecting boundary conditions involve fractional derivatives. The Caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion, since the resulting boundary value problem is not positivity preserving.

Positivity-Preserving Analysis of Numerical Schemes for Ideal Magnetohydrodynamics

Numerical schemes provably preserving the positivity of density and pressure are highly desirable for MHD, but the rigorous positivity-preserving (PP) analysis remains challenging. The difficulties mainly arise from the intrinsic complexity of the MHD equations as well as the indeterminate relation between the PP property and the divergence-free condition on magnetic field. We present the first rigorous PP analysis of conservative schemes with Lax-Friedrichs (LF) flux for ideal MHD. The significant innovation is the discovery of theoretical connection between PP property and a discrete divergence-free (DDF) condition. This connection is established through the generalized LF splitting properties, which are alternatives of the usually-expected LF splitting property that does not hold for ideal MHD. The generalized LF splitting properties involve a number of admissible states strongly coupled by DDF condition, making their derivation very difficult. We derive these properties via a novel equivalent form of the admissible state set and an important inequality skillfully constructed by technical estimates. Rigorous PP analysis is presented for finite volume and discontinuous Galerkin schemes with LF flux on uniform Cartesian meshes. In 1D case, PP property is proved for the first-order scheme with proper numerical viscosity, and also for arbitrarily high-order schemes under conditions accessible by a PP limiter. In 2D case, we show that the DDF condition is necessary and crucial for achieving PP property. It is observed that even slightly violating the proposed DDF condition may cause failure to preserve the positivity of pressure. We prove that the 2D LF type scheme with proper numerical viscosity preserves the positivity and DDF condition. Sufficient conditions are derived for 2D PP high-order schemes, and extension to 3D is discussed. Numerical examples confirm the theoretical findings.