A minimum entropy principle of high order schemes for gas dynamics equations

The entropy solutions of the compressible Euler equations satisfy a minimum principle for the specific entropy (Tadmor in Appl Numer Math 2:211–219, 1986). First order schemes such as Godunov-type and Lax-Friedrichs schemes and the second order kinetic schemes (Khobalatte and Perthame in Math Comput 62:119–131, 1994) also satisfy a discrete minimum entropy principle. In this paper, we show an extension of the positivity-preserving high order schemes for the compressible Euler equations in Zhang and Shu (J Comput Phys 229:8918–8934, 2010) and Zhang et?al. (J Scientific Comput, in press), to enforce the minimum entropy principle for high order finite volume and discontinuous Galerkin (DG) schemes.     

A new fifth order finite difference WENO scheme for Hamilton‐Jacobi equations

In this continuing paper of (Zhu and Qiu, J Comput Phys 318 (2016), 110–121), a new fifth order finite difference weighted essentially non‐oscillatory (WENO) scheme is designed to approximate the viscosity numerical solution of the Hamilton‐Jacobi equations. This new WENO scheme uses the same numbers of spatial nodes as the classical fifth order WENO scheme which is proposed by Jiang and Peng (SIAM J Sci Comput 21 (2000), 2126–2143), and could get less absolute truncation errors and obtain the same order of accuracy in smooth region simultaneously avoiding spurious oscillations nearby discontinuities. Such new WENO scheme is a convex combination of a fourth degree accurate polynomial and two linear polynomials in a WENO type fashion in the spatial reconstruction procedures. The linear weights of three polynomials are artificially set to be any random positive constants with a minor restriction and the new nonlinear weights are proposed for the sake of keeping the accuracy of the scheme in smooth region, avoiding spurious oscillations and keeping sharp discontinuous transitions in nonsmooth region simultaneously. The main advantages of such new WENO scheme comparing with the classical WENO scheme proposed by Jiang and Peng (SIAM J Sci Comput 21 (2000), 2126–2143) are its efficiency, robustness and easy implementation to higher dimensions. Extensive numerical tests are performed to illustrate the capability of the new fifth WENO scheme. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1095–1113, 2017     

A Hybrid Finite Volume Method for Advection Equations and Its Applications in Population Dynamics

We present in this article a very adapted finite volume numerical scheme for transport type‐equation. The scheme is an hybrid one combining an anti‐dissipative method with down‐winding approach for the flux (Després and Lagoutière, C R Acad Sci Paris Sér I Math 328(10) (1999), 939–944; Goudon, Lagoutière, and Tine, Math Method Appl Sci 23(7) (2013), 1177–1215) and an high accurate method as the WENO5 one (Jiang and Shu, J Comput Phys 126 (1996), 202–228). The main goal is to construct a scheme able to capture in exact way the numerical solution of transport type‐equation without artifact like numerical diffusion or without “stairs” like oscillations and this for any regular or discontinuous initial distribution. This kind of numerical hybrid scheme is very suitable when properties on the long term asymptotic behavior of the solution are of central importance in the modeling what is often the case in context of population dynamics where the final distribution of the considered population and its mass preservation relation are required for prediction. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1114–1142, 2017     

An efficient WENO scheme for solving hyperbolic conservation laws

In this paper we propose a new WENO scheme, in which we use a central WENO [G. Capdeville, J. Comput. Phys. 227 (2008) 2977-3014] (CWENO) reconstruction combined with the smoothness indicators introduced in [R. Borges, M. Carmona, B. Costa, W. Sun Don, J. Comput. Phys. 227 (2008) 3191-3211] (IWENO). We use the central-upwind flux [A. Kurganov, S. Noelle, G. Petrova, SIAM J. Sci. Comp. 23 (2001) 707-740] which is simple, universal and efficient. For time integration we use the third order TVD Runge-Kutta scheme. The resulting scheme improves the convergence order at critical points of smooth parts of solution as well as decrease the dissipation near discontinuities. Numerical experiments of the new scheme for one and two-dimensional problems are reported. The results demonstrates that the proposed scheme is superior to the original CWENO and IWENO schemes.     

Notes on convergence of an algebraic multigrid method

The convergence theory for algebraic multigrid (AMG) algorithms proposed in Chang and Huang [Q.S. Chang, Z.H. Huang, Efficient algebraic multigrid algorithms and their convergence, SIAM J. Sci. Comput. 24 (2002) 597–618] is further discussed and a smaller and elegant upper bound is obtained. On the basis of element-free AMGe [V.E. Henson, P.S. Vassilevski, Element-free AMGe: General algorithms for computing interpolation weights in AMG, SIAM J. Sci. Comput. 23(2) (2001) 629–650] we rewrite the interpolation operator for the classical AMG (cAMG), present a uniform expression and then, by introducing a sparse approximate inverse in the Frobenius norm, give a general convergence theorem which is suited for not only cAMG but also AMG for finite elements and element-free AMGe.     

Arbitrary high-order schemes for the linear advection and wave equations: application to hydrodynamics and aeroacoustics

We present here an extension to any order of accuracy of the schemes proposed in Daru and Tenaud [J. Comput. Phys. 193 (2) (2004) 563–594] for the linear advection equation in 1D. Such schemes are then used for a high-order generalization of the Godunov method in the case of the wave equation and the locally linearized Euler equations. To cite this article: S. Del Pino, H. Jourdren, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Some temporal second order difference schemes for fractional wave equations

In this article, motivated by Alikhanov's new work (Alikhanov, J Comput Phys 280 (2015), 424–438), some difference schemes are proposed for both one‐dimensional and two‐dimensional time‐fractional wave equations. The obtained schemes can achieve second‐order numerical accuracy both in time and in space. The unconditional convergence and stability of these schemes in the discrete H¹‐norm are proved by the discrete energy method. The spatial compact difference schemes with the results on the convergence and stability are also presented. In addition, the three‐dimensional problem is briefly mentioned. Numerical examples illustrate the efficiency of the proposed schemes. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 970–1001, 2016     

Analysis of some bivariate non-linear interpolatory subdivision schemes

This paper is devoted to the convergence analysis of a class of bivariate subdivision schemes that can be defined as a specific perturbation of a linear subdivision scheme. We study successively the univariate and bivariate case and apply the analysis to the so called Powerp scheme (Serna and Marquina, J Comput Phys 194:632–658, 2004).     

Analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms

This paper is devoted to the convergence and stability analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms. As soon as a nonlinear scheme can be written as a specific perturbation of a linear and convergent subdivision scheme, we show that if some contractivity properties are satisfied, then stability and convergence can be achieved. This approach is applied to various schemes, which give different new results. More precisely, we study uncentered Lagrange interpolatory linear schemes, WENO scheme (Liu et al., J Comput Phys 115:200–212, 1994), PPH and Power-P schemes (Amat and Liandrat, Appl Comput Harmon Anal 18(2):198–206, 2005; Serna and Marquina, J Comput Phys 194:632–658, 2004) and a nonlinear scheme using local spherical coordinates (Aspert et al., Comput Aided Geom Des 20:165–187, 2003). Finally, a stability proof is given for the multiresolution transform associated to a nonlinear scheme of Marinov et al. (2005).     

Inf‐sup stability of Petrov‐Galerkin immersed finite element methods for one‐dimensional elliptic interface problems

In this article, we analyze the Petrov‐Galerkin immersed finite element method (PG‐IFEM) when applied to one‐dimensional elliptic interface problems. In the PG‐IFEM (T. Hou, X. Wu and Y. Zhang, Commun. Math. Sci., 2 (2004), 185‐205, and S. Hou and X. Liu, J. Comput. Phys., 202 (2005), 411‐445), the classic immersed finite element (IFE) space was taken as the trial space while the conforming linear finite element space was taken as the test space. We first prove the inf‐sup condition of the PG‐IFEM and then show the optimal error estimate in the energy norm. We also show the optimal estimate of the condition number of the stiffness matrix. The results are extended to two dimensional problems in a special case.     

On the convergence of a local third order shock capturing method for hyperbolic conservation laws

The Piecewise Polynomial Harmonic Method (PPHM) is a local third order accurate shock capturing method for hyperbolic conservation laws. In this paper, theoretical stability properties are presented. Using these properties, the convergence of the scheme for scalar conservation laws is obtained. A direct adaptation of these results should be used to derive the convergence of the PHM (Marquina, SIAM J. Sci. Comput. 15(4), 892–915, 1994). Finally, the method is tested in several classical problems in order to explore its numerical behavior.     

WENO schemes with Lax–Wendroff type time discretizations for Hamilton–Jacobi equations

In this paper, a class of weighted essentially non-oscillatory (WENO) schemes with a Lax–Wendroff time discretization procedure, termed WENO-LW schemes, for solving Hamilton–Jacobi equations is presented. This is an alternative method for time discretization to the popular total variation diminishing (TVD) Runge–Kutta time discretizations. We explore the possibility in avoiding the nonlinear weights for part of the procedure, hence reducing the cost but still maintaining non-oscillatory properties for problems with strong discontinuous derivative. As a result, comparing with the original WENO with Runge–Kutta time discretizations schemes (WENO-RK) of Jiang and Peng [G. Jiang, D. Peng, Weighted ENO schemes for Hamilton–Jacobi equations, SIAM J. Sci. Comput. 21 (2000) 2126–2143] for Hamilton–Jacobi equations, the major advantages of WENO-LW schemes are more cost effective for certain problems and their compactness in the reconstruction. Extensive numerical experiments are performed to illustrate the capability of the method.     

A volume‐consistent discrete formulation of aggregation population balance equations

In this paper, a new finite volume scheme for the numerical solution of the pure aggregation population balance equation, or Smoluchowski equation, on non‐uniform meshes is derived. The main feature of the new method is its simple mathematical structure and high accuracy with respect to the number density distribution as well as its moments. The new method is compared with the existing schemes given by Filbet and Laurençot (SIAM J. Sci. Comput., 25 (2004), pp. 2004–2028) and Forestier and Mancini (SIAM J. Sci. Comput., 34 (2012), pp. B840–B860) for selected benchmark problems. It is shown that the new scheme preserves all the advantages of a conventional finite volume scheme and predicts higher‐order moments as well as number density distribution with high accuracy. Copyright © 2015 John Wiley & Sons, Ltd.     

An efficiently computed lower bound on the number of recombinations in phylogenetic networks: Theory and empirical study

Phylogenetic networks are models of sequence evolution that go beyond trees, allowing biological operations that are not tree-like. One of the most important biological operations is recombination between two sequences. An established problem [J. Hein, Reconstructing evolution of sequences subject to recombination using parsimony, Math. Biosci. 98 (1990) 185–200; J. Hein, A heuristic method to reconstruct the history of sequences subject to recombination, J. Molecular Evoluation 36 (1993) 396–405; Y. Song, J. Hein, Parsimonious reconstruction of sequence evolution and haplotype blocks: finding the minimum number of recombination events, in: Proceedings of 2003 Workshop on Algorithms in Bioinformatics, Berlin, Germany, 2003, Lecture Notes in Computer Science, Springer, Berlin; Y. Song, J. Hein, On the minimum number of recombination events in the evolutionary history of DNA sequences, J. Math. Biol. 48 (2003) 160–186; L. Wang, K. Zhang, L. Zhang, Perfect phylogenetic networks with recombination, J. Comput. Biol. 8 (2001) 69–78; S.R. Myers, R.C. Griffiths, Bounds on the minimum number of recombination events in a sample history, Genetics 163 (2003) 375–394; V. Bafna, V. Bansal, Improved recombination lower bounds for haplotype data, in: Proceedings of RECOMB, 2005; Y. Song, Y. Wu, D. Gusfield, Efficient computation of close lower and upper bounds on the minimum number of needed recombinations in the evolution of biological sequences, Bioinformatics 21 (2005) i413–i422. Bioinformatics (Suppl. 1), Proceedings of ISMB, 2005, D. Gusfield, S. Eddhu, C. Langley, Optimal, efficient reconstruction of phylogenetic networks with constrained recombination, J. Bioinform. Comput. Biol. 2(1) (2004) 173–213; D. Gusfield, Optimal, efficient reconstruction of root-unknown phylogenetic networks with constrained and structured recombination, J. Comput. Systems Sci. 70 (2005) 381–398] is to find a phylogenetic network that derives an input set of sequences, minimizing the number of recombinations used. No efficient, general algorithm is known for this problem. Several papers consider the problem of computing a lower bound on the number of recombinations needed. In this paper we establish a new, efficiently computed lower bound. This result is useful in methods to estimate the number of needed recombinations, and also to prove the optimality of algorithms for constructing phylogenetic networks under certain conditions [D. Gusfield, S. Eddhu, C. Langley, Optimal, efficient reconstruction of phylogenetic networks with constrained recombination, J. Bioinform. Comput. Biol. 2(1) (2004) 173–213; D. Gusfield, Optimal, efficient reconstruction of root-unknown phylogenetic networks with constrained and structured recombination, J. Comput. Systems Sci. 70 (2005) 381–398; D. Gusfield, Optimal, efficient reconstruction of root-unknown phylogenetic networks with constrained recombination, Technical Report, Department of Computer Science, University of California, Davis, CA, 2004]. The lower bound is based on a structural, combinatorial insight, using only the site conflicts and incompatibilities, and hence it is fundamental and applicable to many biological phenomena other than recombination, for example, when gene conversions or recurrent or back mutations or cross-species hybridizations cause the phylogenetic history to deviate from a tree structure. In addition to establishing the bound, we examine its use in more complex lower bound methods, and compare the bounds obtained to those obtained by other established lower bound methods.     

Recent advances in linear analysis of convergence for splittings for solving ODE problems

In the nineties, Van der Houwen et al. (see, e.g., [P.J. van der Houwen, B.P. Sommeijer, J.J. de Swart, Parallel predictor–corrector methods, J. Comput. Appl. Math. 66 (1996) 53–71; P.J. van der Houwen, J.J.B. de Swart, Triangularly implicit iteration methods for ODE-IVP solvers, SIAM J. Sci. Comput. 18 (1997) 41–55; P.J. van der Houwen, J.J.B. de Swart, Parallel linear system solvers for Runge–Kutta methods, Adv. Comput. Math. 7 (1–2) (1997) 157–181]) introduced a linear analysis of convergence for studying the properties of the iterative solution of the discrete problems generated by implicit methods for ODEs. This linear convergence analysis is here recalled and completed, in order to provide a useful quantitative tool for the analysis of splittings for solving such discrete problems. Indeed, this tool, in its complete form, has been actively used when developing the computational codes BiM and BiMD [L. Brugnano, C. Magherini, The BiM code for the numerical solution of ODEs, J. Comput. Appl. Math. 164–165 (2004) 145–158. Code available at: http://www.math.unifi.it/~brugnano/BiM/index.html; L. Brugnano, C. Magherini, F. Mugnai, Blended implicit methods for the numerical solution of DAE problems, J. Comput. Appl. Math. 189 (2006) 34–50]. Moreover, the framework is extended for the case of special second order problems. Examples of application, aimed to compare different iterative procedures, are also presented.     

Hybrid WENO schemes with different indicators on curvilinear grids

In {J. Comput. Phys. 229 (2010) 8105-8129}, we studied hybrid weighted essentially non-oscillatory (WENO) schemes with different indicators for hyperbolic conservation laws on uniform grids for Cartesian domains. In this paper, we extend the schemes to solve two-dimensional systems of hyperbolic conservation laws on curvilinear grids for non-Cartesian domains. Our goal is to obtain similar advantageous properties as those of the hybrid WENO schemes on uniform grids for Cartesian domains. Extensive numerical results strongly support that the hybrid WENO schemes with discontinuity indicators on curvilinear grids can also save considerably on computational cost in contrast to the pure WENO schemes. They also maintain the essentially non-oscillatory property for general solutions with discontinuities and keep the sharp shock transition.     

Adiabatic Non-Equilibrium Steady States in the Partition Free Approach

Consider a small sample coupled to a finite number of leads and assume that the total (continuous) system is at thermal equilibrium in the remote past. We construct a non-equilibrium steady state (NESS) by adiabatically turning on an electrical bias between the leads. The main mathematical challenge is to show that certain adiabatic wave operators exist and to identify their strong limit when the adiabatic parameter tends to zero. Our NESS is different from, though closely related with the NESS provided by the Jakšić–Pillet–Ruelle approach. Thus we partly settle a question asked by Caroli et al. (J. Phys. C Solid State Phys. 4(8):916–929, 1971) regarding the (non)equivalence between the partitioned and partition-free approaches.     

Convergence Analysis of a Numerical Scheme for the Porous Medium Equation by an Energetic Variational Approach

The porous medium equation (PME)is a typical nonlinear degenerate parabolic equation. We have studied numerical methods for PME by an energetic variational approach in [C. Duan et al., J. Comput. Phys., 385 (2019), pp. 13–32], where the trajectory equation can be obtained and two numerical schemes have been developed based on different dissipative energy laws. It is also proved that the nonlinear scheme, based on $f$log $f$ as the total energy form of the dissipative law, is uniquely solvable on an admissible convex set and preserves the corresponding discrete dissipation law. Moreover, under certain smoothness assumption, we have also obtained the second order convergence in space and the first order convergence in time for the scheme. In this paper, we provide a rigorous proof of the error estimate by a careful higher order asymptotic expansion and two step error estimates. The latter technique contains a rough estimate to control the highly nonlinear term in a discrete $W$^1,∞norm and a refined estimate is applied to derive the optimal error order.     

On positive solutions in a phytoplankton–nutrient model

On the basis of an application from aquatic ecology, we discuss the behaviour of the widely used time integration package VODE by Brown et al. (SIAM J. Sci. Statist. Comput. 10 (1989) 1038). When used in a default setting this code smoothly produces a negative steady state solution, which is not realistic in this application.     

Convergence of a multigrid method for the controllability of a 1-d wave equation

We consider the problem of computing numerically the boundary control for the wave equation. It is by now well known that, due to high frequency spurious oscillations, numerical instabilities occur and may led to the failure of convergence of some apparently natural numerical algorithms. Several remedies have been proposed in the literature to compensate this fact: Tychonoff regularization, Fourier filtering, mixed finite elements,… In this Note we prove that the two-grid method proposed by Glowinski (J. Comput. Phys. 103 (2) (1992) 189–221) does indeed provide a convergent algorithm. This is done in the context of the finite-difference semi-discrete approximation of the 1-d wave equation. To cite this article: M. Negreanu, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 338 (2004).