Curl Constraint-Preserving Reconstruction and the Guidance it Gives for Mimetic Scheme Design

Several important PDE systems,like magnetohydrodynamics and computational electrodynamics,are known to support involutions where the divergence of a vector field evolves in divergence-free or divergence constraint-preserving fashion.Recently,new classes of PDE systems have emerged for hyperelasticity,compressible multiphase flows,so-called firstorder reductions of the Einstein field equations,or a novel first-order hyperbolic reformulation of Schrödinger’s equation,to name a few,where the involution in the PDE supports curl-free or curl constraint-preserving evolution of a vector field.We study the problem of curl constraint-preserving reconstruction as it pertains to the design of mimetic finite volume(FV)WENO-like schemes for PDEs that support a curl-preserving involution.(Some insights into discontinuous Galerkin(DG)schemes are also drawn,though that is not the prime focus of this paper.)This is done for two-and three-dimensional structured mesh problems where we deliver closed form expressions for the reconstruction.The importance of multidimensional Riemann solvers in facilitating the design of such schemes is also documented.In two dimensions,a von Neumann analysis of structure-preserving WENOlike schemes that mimetically satisfy the curl constraints,is also presented.It shows the tremendous value of higher order WENO-like schemes in minimizing dissipation and dispersion for this class of problems.Numerical results are also presented to show that the edge-centered curl-preserving(ECCP)schemes meet their design accuracy.This paper is the first paper that invents non-linearly hybridized curl-preserving reconstruction and integrates it with higher order Godunov philosophy.By its very design,this paper is,therefore,intended to be forward-looking and to set the stage for future work on curl involution-constrained PDEs.

Von Neumann Stability Analysis of DG-Like and PNPM-Like Schemes for PDEs with Globally Curl-Preserving Evolution of Vector Fields

This paper examines a class of involution-constrained PDEs where some part of the PDE system evolves a vector field whose curl remains zero or grows in proportion to specified source terms.Such PDEs are referred to as curl-free or curl-preserving,respectively.They arise very frequently in equations for hyperelasticity and compressible multiphase flow,in certain formulations of general relativity and in the numerical solution of Schrödinger’s equation.Experience has shown that if nothing special is done to account for the curl-preserving vector field,it can blow up in a finite amount of simulation time.In this paper,we catalogue a class of DG-like schemes for such PDEs.To retain the globally curl-free or curl-preserving constraints,the components of the vector field,as well as their higher moments,must be collocated at the edges of the mesh.They are updated using potentials collocated at the vertices of the mesh.The resulting schemes:(i)do not blow up even after very long integration times,(ii)do not need any special cleaning treatment,(iii)can oper-ate with large explicit timesteps,(iv)do not require the solution of an elliptic system and(v)can be extended to higher orders using DG-like methods.The methods rely on a spe-cial curl-preserving reconstruction and they also rely on multidimensional upwinding.The Galerkin projection,highly crucial to the design of a DG method,is now conducted at the edges of the mesh and yields a weak form update that uses potentials obtained at the verti-ces of the mesh with the help of a multidimensional Riemann solver.A von Neumann sta-bility analysis of the curl-preserving methods is conducted and the limiting CFL numbers of this entire family of methods are catalogued in this work.The stability analysis confirms that with the increasing order of accuracy,our novel curl-free methods have superlative phase accuracy while substantially reducing dissipation.We also show that PNPM-like methods,which only evolve the lower moments while reconstructing the higher moments,retain much of the excellent wave propagation characteristics of the DG-like methods while offering a much larger CFL number and lower computational complexity.The quadratic energy preservation of these methods is also shown to be excellent,especially at higher orders.The methods are also shown to be curl-preserving over long integration times.

KINETIC FUNCTIONS IN MAGNETOHYDRODYNAMICS WITH RESISTIVITY AND HALL EFFECT

We consider a nonlinear hyperbolic system of two conservation laws which arises in ideal magnetohydrodynamics and includes second-order terms accounting for magnetic resistivity and Hall effect. We show that the initial value problem for this model may lead to solutions exhibiting complex wave structures, including undercompressive nonclassical shock waves. We investigate numerically the subtle competition that takes place between the hyperbolic, diffusive, and dispersive parts of the system. Following Abeyratne, Knowles, LeFloch, and Truskinovsky, who studied similar questions arising in fluid and solid flows, we determine the associated kinetic function which characterizes the dynamics of undereompressive shocks driven by resistivity and Hall effect. To this end, we design a new class of ＂schemes with eontroled dissipation＂, following recent work by LeFloch and Mohammadian. It is now recognized that the equivalent equation associated with a scheme provides a guideline to design schemes that capture physically relevant, nonclassical shocks. We propose a new class of schemes based on high-order entropy conservative, finite differences for the hyperbolic flux, and high-order central differences for the resistivity and Hall terms. These schemes are tested for several regimes of （co-planar or not） initial data and parameter values, and allow us to analyze the properties of nonclassical shocks and establish the existence of monotone kinetic functions in magnetohydrodynamics.

PHYSICS INFORMED NEURAL NETWORKS (PINNs) FOR APPROXIMATING NONLINEAR DISPERSIVE PDEs

We propose a novel algorithm,based on physics-informed neural networks(PINNs)to efficiently approximate solutions of nonlinear dispersive PDEs such as the KdV-Kawahara,Camassa-Holm and Benjamin-Ono equations.The stability of solutions of these dispersive PDEs is leveraged to prove rigorous bounds on the resulting error.We present several numerical experiments to demonstrate that PINNs can approximate solutions of these dispersive PDEs very accurately.

FAST SPECTRAL GALERKIN METHOD FOR LOGARITHMIC SINGULAR EQUATIONS ON A SEGMENT＊

We present a fast Galerkin spectral method to solve logarithmic singular equations on segments. The proposed method uses weighted first-kind Chebyshev polynomials. Conver- gence rates of several orders are obtained for fractional Sobolev spaces H^-1/2 （or H00^-l/2）. Main tools are the approximation properties of the discretization basis, the construction of a suitable Hilbert scale for weighted L2-spaces and local regularity estimates. Numerical experiments are provided to validate our claims,

Approximate Riemann Solvers and Robust High-Order Finite Volume Schemes for Multi-Dimensional Ideal MHD Equations

We design stable and high-order accurate finite volume schemes for the ideal MHD equations in multi-dimensions.We obtain excellent numerical stability due to some new elements in the algorithm.The schemes are based on three-and five-wave approximate Riemann solvers of the HLL-type,with the novelty that we allow a varying normal magnetic field.This is achieved by considering the semiconservative Godunov-Powell form of the MHD equations.We show that it is important to discretize the Godunov-Powell source term in the right way,and that the HLL-type solvers naturally provide a stable upwind discretization.Second-order versions of the ENO-and WENO-type reconstructions are proposed,together with precise modifications necessary to preserve positive pressure and density.Extending the discrete source term to second order while maintaining stability requires non-standard techniques,which we present.The first-and second-order schemes are tested on a suite of numerical experiments demonstrating impressive numerical resolution as well as stability,even on very fine meshes.

Second-Kind Boundary Integral Equations for Scattering at Composite Partly Impenetrable Objects

We consider acoustic scattering of time-harmonic waves at objects composed of several homogeneous parts.Some of those may be impenetrable,giving rise to Dirichlet boundary conditions on their surfaces.We start from the recent secondkind boundary integral approach of[X.Claeys,and R.Hiptmair,and E.Spindler.A second-kind Galerkin boundary element method for scattering at composite objects.BIT Numerical Mathematics,55(1):33-57,2015]for pure transmission problems and extend it to settings with essential boundary conditions.Based on so-called global multipotentials,we derive variational second-kind boundary integral equations posed in L^(2)(S),where S denotes the union of material interfaces.To suppress spurious resonances,we introduce a combined-field version(CFIE)of our new method.Thorough numerical tests highlight the low andmesh-independent condition numbers of Galerkin matrices obtained with discontinuous piecewise polynomial boundary element spaces.They also confirm competitive accuracy of the numerical solution in comparison with the widely used first-kind single-trace approach.

Hyperbolic Moment Equations in Kinetic Gas Theory Based on Multi-Variate Pearson-IV-Distributions

In this paper we develop a new closure theory for moment approximationsin kinetic gas theory and derive hyperbolic moment equations for 13 fluid variablesincluding stress and heat flux. Classical equations have either restricted hyperbolicity regions like Grad’s moment equations or fail to include higher moments in apractical way like the entropy maximization approach. The new closure is based onPearson-Type-IV distributions which reduce to Maxwellians in equilibrium, but allowanisotropies and skewness in non-equilibrium. The closure relations are essentiallyexplicit and easy to evaluate. Hyperbolicity is shown numerically for a large range ofvalues. Numerical solutions of Riemann problems demonstrate the capability of thenew equations to handle strong non-equilibrium.