On unconditionally positivity preserving DG schemes for shallow water flows with shock capturing by adaptive filtering procedures

In this work, we present an unconditionally positivity preserving implicit time integration scheme for the DG method applied to shallow water flows. For locally refined grids with very small elements, the ODE resulting from space discretization is stiff and requires implicit or partially implicit time stepping. However, for simulations including wetting and drying or regions with small water height, severe time step restrictions have to be imposed due to positivity preservation. Nevertheless, as implicit time stepping demands a significant amount of computational time in order to solve a large system of nonlinear equations for each time step we need large time steps to obtain an efficient scheme. In this context, we propose a novel approach to the strategy of positivity preservation. This new technique is based on the so-called Patankar trick and guarantees non-negativity of the water height for any time step size while still preserving conservativity. Due to this modification, the implicit scheme can take full advantage of larger time steps and is therefore able to beat explicit time stepping in terms of CPU time. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Real polynomials: A test for non-global non-negativity and non-global positivity

Tests for non-global non-negativity and non-global positivity of a real polynomial in m real variables are described. Each test consists of two phases. In the first phase, non-global non-negativity or non-global positivity of the polynomial under consideration is equated to (global) non-negativity or (global) positivity of a derived polynomial. In the second phase, a test for non-negativity or positivity is invoked. The class of non-global domains that can be transformed to global domains is considered and illustrated.     

Application of spectral filtering to discontinuous Galerkin methods on triangulations

We adapt the spectral viscosity (SV) formulation implemented as a modal filter to a discontinuous Galerkin (DG) method solving hyperbolic conservation laws on triangular grids. The connection between SV and spectral filtering, which is undertaken for the first time in the context of DG methods on unstructured grids, allows to specify conditions on the filter strength regarding time step choice and mesh refinement. A crucial advantage of this novel damping strategy is its low computational cost. We furthermore obtain new error bounds for filtered Dubiner expansions of smooth functions. While high order accuracy with respect to the polynomial degree N is proven for the filtering procedure in this case, an adaptive application is proposed to retain the high spatial approximation order. Although spectral filtering stabilizes the scheme, it leaves weaker oscillations. Therefore, as a postprocessing step, we apply the image processing technique of digital total variation (DTV) filtering in the new context of DG solutions and prove conservativity in the limit for this filtering procedure. Numerical experiments for scalar conservation laws confirm the designed order of accuracy of the DG scheme with adaptive modal filtering for polynomial degrees up to 8 and the viability of spectral and DTV filtering in case of shocks. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011     

A Study on CFL Conditions for the DG Solution of Conservation Laws on Adaptive Moving Meshes

The selection of time step plays a crucial role in improving stability and efficiency in the Discontinuous Galerkin (DG) solution of hyperbolic conservation laws on adaptive moving meshes that typically employs explicit stepping. A commonly used selection of time step is a direct extension based on Courant-Friedrichs-Levy (CFL) conditions established for fixed and uniform meshes. In this work, we provide a mathematical justification for those time step selection strategies used in practical adaptive DG computations. A stability analysis is presented for a moving mesh DG method for linear scalar conservation laws. Based on the analysis, a new selection strategy of the time step is proposed, which takes into consideration the coupling of the $α$-function (that is related to the eigenvalues of the Jacobian matrix of the flux and the mesh movement velocity) and the heights of the mesh elements. The analysis also suggests several stable combinations of the choices of the $α$-function in the numerical scheme and in the time step selection. Numerical results obtained with a moving mesh DG method for Burgers' and Euler equations are presented. For comparison purpose, numerical results obtained with an error-based time step-size selection strategy are also given.     

On K-Positivity Properties of Time-Invariant Linear Systems Subject to Discrete Point Lags

This paper discusses non-negativity and positivity concepts and related properties for the state and output trajectory solutions of dynamic linear time-invariant systems described by functional differential equations subject to point time-delays. The various non-negativity and positivity introduced hierarchically from the weakest one to the strongest one while separating the corresponding properties when applied to the state space or to the output space as well as for the zero-initial state or zero-input responses. The formulation is developed by defining cones for the input, state and output spaces of the dynamic system.     

A discontinuous Galerkin method for a new class of Green–Naghdi equations on simplicial unstructured meshes

In this paper, we introduce a discontinuous Finite Element formulation on simplicial unstructured meshes for the study of free surface flows based on the fully nonlinear and weakly dispersive Green-Naghdi equations. Working with a new class of asymptotically equivalent equations, which have a simplified analytical structure, we consider a decoupling strategy: we approximate the solutions of the classical shallow water equations supplemented with a source term globally accounting for the non-hydrostatic effects and we show that this source term can be computed through the resolution of scalar elliptic second-order sub-problems. The assets of the proposed discrete formulation are: (i) the handling of arbitrary unstructured simplicial meshes, (ii) an arbitrary order of approximation in space, (iii) the exact preservation of the motionless steady states, (iv) the preservation of the water height positivity, (v) a simple way to enhance any numerical code based on the nonlinear shallow water equations. The resulting numerical model is validated through several benchmarks involving nonlinear wave transformations and run-up over complex topographies.     

Preserving positivity for hyperbolic PDEs using variable-order finite elements with bounded polynomials

The positivity preserving approach of Berzins is generalized by using a derivation based on bounded polynomial approximations and order selection. The approach is extended from the B-spline based methods used previously to the use of more conventional continuous Galerkin elements. The conditions relating to positivity preservation are considered and a numerical example used to demonstrate the performance of the method on a model advection equation problem.     

A kinetic energy preserving DG scheme applied to mechanical FSI

In the context of mechanical fluid-structure interaction (FSI) comprising moving or deforming structures, fluid discretizations need to cope with time-dependent fluid domains and resulting grid deformations in addition to the general challenges regarding e.g. boundary layers and turbulent phenomena. Recent approaches in the simulation of compressible turbulent flow are based on so-called split forms of conservation laws to guarantee the preservation of secondary physical quantities such as kinetic energy. For the simulation of turbulent flows, this often leads to a better representation of the kinetic energy spectrum. Initially, kinetic energy preserving(KEP) DG schemes have been constructed on Gauss-Legendre-Lobatto(GLL) nodes containing the interval end points, however, KEP DG schemes based on the classical Gauss-Legendre(GL) nodes are potentially more accurate and may be also more efficient than its GLL variant for certain applications. In this work, the KEP-DG schemes both on GL and GLL nodes are applied to the classical moving piston test case via an ALE formulation on moving fluid grids showing a more accurate frequency representation of the structure displacement in case of GLL nodes. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Unconditionally Positivity Preserving and Conservative Methods for Systems of Advection-Diffusion-Reaction Equations

An unconditionally positivity preserving finite difference scheme (UPFD) for systems of advection-diffusion-reaction equations with non-linear reaction terms is proposed. A modified Patankar approach is employed with respect to the reaction part in order to ensure both conservativity and positivity without any additional constraints on the time step size. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A flux-corrected transport algorithm for handling the close-packing limit in dense suspensions

Convection of a scalar quantity by a compressible velocity field may give rise to unbounded solutions or nonphysical overshoots at the continuous and discrete level. In this paper, we are concerned with solving continuity equations that govern the evolution of volume fractions in Eulerian models of disperse two-phase flows. An implicit Galerkin finite element approximation is equipped with a flux limiter for the convective terms. The fully multidimensional limiting strategy is based on a flux-corrected transport (FCT) algorithm. This nonlinear high-resolution scheme satisfies a discrete maximum principle for divergence-free velocities and ensures positivity preservation for arbitrary velocity fields. To enforce an upper bound that corresponds to the maximum-packing limit, an FCT-like overshoot limiter is applied to the converged convective fluxes at the end of each time step. This postprocessing step imposes an additional physical constraint on the numerical solution to the unconstrained mathematical model. Numerical results for 2D implosion problems illustrate the performance of the proposed limiting procedure.     

Obtuse cones in Hilbert spaces and applications to partial differential equations

The positive cone K in a partially ordered Hilbert space is said to be obtuse with respect to the inner product if the dual cone $\text{K}{}^1\text{? K}$. Obtuseness of cones with respect to non-symmetric bilinear forms is also defined and characterized. These results are applied to the generalized Sobolev space associated with an elliptic boundary value problem, in particular to the question of determining the non-negativity of the Green's function. A notion of strict obtuseness is defined, characterized and applied to the question of strict positivity of the Green's function. Applications to positivity preserving semi-groups are also given.     

Variational multirate integration of constrained dynamics

Mechanical systems with dynamics on varying time scales, in particular those including highly oscillatory motion, impose challenging questions for numerical integration schemes. Tiny step sizes are required to guarantee a stable integration of the fast frequencies. However, for the simulation of the slow dynamics, integration with a larger time step is accurate enough. Small time steps increase integration times unnecessarily, especially for costly function evaluations. For systems comprising fast and slow dynamics, multirate methods integrate the slow part of the system with a relatively large step size while the fast part is integrated with a small time step. Main challenges are the identification of fast and slow parts (e.g. by separating the energy or by distinguishing sets of variables), the synchronisation of their dynamics and in particular the treatment of mixed parts that often appear when fast and slow dynamics are coupled by constraints. In this contribution, a multirate integrator is derived in closed form via a discrete variational principle on a time grid consisting of macro and micro time nodes. Variational integrators (based on a discrete version of Hamilton's principle) lead to symplectic and momentum preserving integration schemes that also exhibit good energy behavior. The resulting multirate variational integrator has the same preservation properties. An example demonstrates the performance of the multirate integrator for constrained multibody dynamics. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

解非负约束图像去模糊问题的积极集方法

研究非负约束全变分图像去模糊问题,提出了一个基于增广拉格朗日方法的积极集方法,并证明了该方法在有限步内可求解,进一步推出该方法等价于解非光滑方程组的半光滑牛顿法.     

A vertex‐centered and positivity‐preserving scheme for anisotropic diffusion equations on general polyhedral meshes

We propose a new nonlinear positivity‐preserving finite volume scheme for anisotropic diffusion problems on general polyhedral meshes with possibly nonplanar faces. The scheme is a vertex‐centered one where the edge‐centered, face‐centered, and cell‐centered unknowns are treated as auxiliary ones that can be computed by simple second‐order and positivity‐preserving interpolation algorithms. Different from most existing positivity‐preserving schemes, the presented scheme is based on a special nonlinear two‐point flux approximation that has a fixed stencil and does not require the convex decomposition of the co‐normal. More interesting is that the flux discretization is actually performed on a fixed tetrahedral subcell of the primary cell, which makes the scheme very easy to be implemented on polyhedral meshes with star‐shaped cells. Moreover, it is suitable for polyhedral meshes with nonplanar faces, and it does not suffer the so‐called numerical heat‐barrier issue. The truncation error is analyzed rigorously, while the Picard method and its Anderson acceleration are used for the solution of the resulting nonlinear system. Numerical experiments are also provided to demonstrate the second‐order accuracy and well positivity of the numerical solution for heterogeneous and anisotropic diffusion problems on severely distorted grids.     

A parallel finite volume scheme preserving positivity for diffusion equation on distorted meshes

Parallel domain decomposition methods are natural and efficient for solving the implicity schemes of diffusion equations on massive parallel computer systems. A finite volume scheme preserving positivity is essential for getting accurate numerical solutions of diffusion equations and ensuring the numerical solutions with physical meaning. We call their combination as a parallel finite volume scheme preserving positivity, and construct such a scheme for diffusion equation on distorted meshes. The basic procedure of constructing the parallel finite volume scheme is based on the domain decomposition method with the prediction‐correction technique at the interface of subdomains: First, we predict the values on each inner interface of subdomains partitioned by the domain decomposition. Second, we compute the values in each subdomain using a finite volume scheme preserving positivity. Third, we correct the values on each inner interface using the finite volume scheme preserving positivity. The resulting scheme has intrinsic parallelism, and needs only local communication among neighboring processors. Numerical results are presented to show the performance of our schemes, such as accuracy, stability, positivity, and parallel speedup.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2159–2178, 2017     

On positivity preserving finite volume schemes for Euler equations

Summary. We consider the positivity preserving property of first and higher order finite volume schemes for one and two dimensional Euler equations of gas dynamics. A general framework is established which shows the positivity of density and pressure whenever the underlying one dimensional first order building block based on an exact or approximate Riemann solver and the reconstruction are both positivity preserving. Appropriate limitation to achieve a high order positivity preserving reconstruction is described. Received May 20, 1994     

Riesz-Haviland criterion for incomplete data

The aim of this paper is to analyze a “support-free” version of the Riesz-Haviland theorem proved recently by the present authors, which characterizes truncations of the complex moment problem via positivity condition on appropriate families of polynomials in z and . The attention is focused on modifications of the positivity condition as well as the assumption on admissible truncations. The former results in truncations for which the corresponding “support-free” Riesz-Haviland condition locates a representing measure on the distinguished subset of the complex plane, while the latter effects a non-integral variant of the Riesz-Haviland theorem.     

On Green’s functions for positive,elliptic differential operators on closed,Riemannian manifolds and some consequences for semi-linear PDE

In this article, we establish the non-negativity of Green’s functions for a class of elliptic differential operators on closed, Riemannian manifolds. The method used is the calculus of variations with a unilateral constraint. Consequences of this result are then explored, with a maximum principle-type result established for a broad class of semi-linear elliptic PDE as a result. Finally, a result regarding the positivity of the principal eigenfunction of an operator in this class is proven as well.     

A Mickens-type monotone discretization for bounded travelling-wave solutions of a Burgers–Fisher partial differential equationa*

We present a nonlinear method to approximate solutions of a Burgers–Huxley equation with generalized advection factor and logistic reaction. The equation under investigation possesses travelling-wave solutions that are temporally and spatially monotone functions; the travelling-wave fronts considered are bounded and connect asymptotically the stationary solutions of the model. For the linear regime, the method is consistent of first order in time and second order in space. In the nonlinear scenario, we investigate conditions under which bounded initial profiles evolve into bounded new approximations. The main results report on parametric conditions that guarantee the boundedness, the positivity and the monotonicity preservation of the method. As a consequence, our recursive method is capable of preserving the temporal and the spatial monotonicity of the solutions. We provide simulations that show that, indeed, our technique preserves the positivity, the boundedness and the temporal and spatial monotonicity of solutions.     

Central-upwind scheme for shallow water equations with discontinuous bottom topography

Finite-volume central-upwind schemes for shallow water equations were proposed in [A. Kurganov and G. Petrova, Commun. Math. Sci., 5 (2007), 133–160]. These schemes are capable of maintaining “lake-at-rest” steady states and preserving the positivity of the computed water depth. The well-balanced and positivity preserving features of the central-upwind schemes are achieved, in particular, by using continuous piecewise linear interpolation of the bottom topography function. However, when the bottom function is discontinuous or a model with a moving bottom topography is studied, the continuous piecewise linear approximationmay not be sufficiently accurate and robust.