High‐order asymptotic‐preserving schemes for linear systems: Application to the Goldstein–Taylor equations

In this work, we address the numerical approximation of linear systems with possibly stiff source terms which induce an asymptotic diffusion limit. More precisely, we are interested in the design of high‐order asymptotic‐preserving schemes. Our approach is based on a very simple modification of the numerical flux associated with the usual HLL scheme. This alteration can be understood as a numerical diffusion reduction technique and allows to capture the correct asymptotic behavior in the diffusion limit and to consider uniformly high‐order extensions. We more specifically consider the case of the Goldstein–Taylor model but the overall approach is shown to be easily adapted to more general systems.     

Grid approximation of singularly perturbed parabolic reaction-diffusion equations on large domains with respect to the space and time variables

In an unbounded (with respect to x and t) domain (and in domains that can be arbitrarily large), an initial-boundary value problem for singularly perturbed parabolic reaction-diffusion equations with the perturbation parameter ε² multiplying the higher order derivative is considered. The parameter ε takes arbitrary values in the half-open interval (0, 1]. To solve this problem, difference schemes on grids with an infinite number of nodes (formal difference schemes) are constructed that converge ε-uniformly in the entire unbounded domain. To construct these schemes, the classical grid approximations of the problem on the grids that are refined in the boundary layer are used. Schemes on grids with a finite number of nodes (constructive difference schemes) are also constructed for the problem under examination. These schemes converge for fixed values of ε in the prescribed bounded subdomains that can expand as the number of grid points increases. As ε → 0, the accuracy of the solution provided by such schemes generally deteriorates and the size of the subdomains decreases. Using the condensing grid method, constructive difference schemes that converge ε-uniformly are constructed. In these schemes, the approximation accuracy and the size of the prescribed subdomains (where the schemes are convergent) are independent of ε and the subdomains may expand as the number of nodes in the underlying grids increases.     

High‐resolution monotone schemes based on quasi‐characteristics technique

In this article, we consider a new technique that allows us to overcome the well‐known restriction of Godunov's theorem. According to Godunov's theorem, a second‐order explicit monotone scheme does not exist. The techniques in the construction of high‐resolution schemes with monotone properties near the discontinuities of the solution lie in choosing of one of two high‐resolution numerical solutions computed on different stencils. The criterion for choosing the final solution is proposed. Results of numerical tests that compare with the exact solution and with the numerical solution obtained by the first‐order monotone scheme are presented. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 262–276, 2001     

An efficient splitting technique for two‐layer shallow‐water model

We consider the numerical approximation of the weak solutions of the two‐layer shallow‐water equations. The model under consideration is made of two usual one‐layer shallow‐water model coupled by nonconservative products. Because of the nonconservative products of the system, which couple both one‐layer shallow‐water subsystems, the usual numerical methods have to consider the full model. Of course, uncoupled numerical techniques, just involving finite volume schemes for the basic shallow‐water equations, are very attractive since they are very easy to implement and they are costless. Recently, a stable layer splitting technique was introduced [Bouchut and Morales de Luna, M2AN Math Model Numer Anal 42 (2008), 683–698]. In the same spirit, we exhibit new splitting technique, which is proved to be well balanced and non‐negative preserving. The main benefit issuing from the here derived uncoupled method is the ability to correctly approximate the solution of very severe benchmarks. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1396–1423, 2015     

Time discretizations for numerical optimisation of hyperbolic problems

We consider a class of numerical schemes for optimal control problems of hyperbolic conservation laws. We focus on finite-volume schemes using relaxation as a numerical approach to the optimality system. In particular, we study the arising numerical schemes for the adjoint equation and derive necessary conditions on the time integrator. We show that the resulting schemes are in particular asymptotic preserving for both, the adjoint and forward equation. We furthermore prove that higher-order time-integrator yields suitable Runge-Kutta schemes. The discussion includes the numerically interesting zero relaxation case.     

Asymptotic solution to the restricted three‐body problem with a mass point moving near a small‐mass body

The motion of a mass point in the gravitational field of two bodies traveling along arbitrary orbits about their center of mass is considered. The mass ratio of the two bodies is equal to ε?1. When the mass point passes close to the smaller mass, the character of its trajectory changes abruptly, and the trajectory asymptotics as ε→0 is complex. The uniform asymptotic expansion of the entire trajectory up to any power of ε is constructed and justified. In particular, an algorithm is presented for finding the limiting turning angle of the trajectory after the mass point passes a neighbourhood of the smaller mass. Copyright © 2010 John Wiley & Sons, Ltd.     

On splitting‐based mass and total energy conserving arbitrary order shallow‐water schemes

A new method for numerical solution to the shallow‐water equations is suggested. The method allows constructing a family of finite difference schemes of different approximation order that conserve the mass and the total energy. Our approach is based on the method of splitting, and unlike others it permits to derive conservative numerical schemes after discretizing all the partial derivatives, both spatial and temporal. The schemes thus appear to be fully discrete, both in time and in space. Besides, due to a simple structure of the matrices appeared therewith, the method provides essential benefits in the computational cost of solution and is easy‐to‐implement in the Cartesian and spherical geometries. Numerical results confirm functionality and efficiency of the developed method. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

Conservative semi‐Lagrangian finite volume schemes

Semi‐Lagrangian finite volume schemes for the numerical approximation of linear advection equations are presented. These schemes are constructed so that the conservation properties are preserved by the numerical approximation. This is achieved using an interpolation procedure based on area‐weighting. Numerical results are presented illustrating some of the features of these schemes. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:403–425, 2001     

Indecomposable quasi‐characteristics scheme on pyramidal stencil and its application for numerical simulation of two‐phase flows through heterogeneous porous medium

A new high‐resolution indecomposable quasi‐characteristics scheme with monotone properties based on pyramidal stencil is considered. This scheme is based on consideration of two high‐resolution numerical schemes approximated governing equations on the pyramidal stencil with different kinds of dispersion terms approximation. Two numerical solutions obtained by these schemes are analyzed, and the final solution is chosen according to the special criterion to provide the monotone properties in regions where discontinuities of solutions could arise. This technique allows to construct the high‐order monotone solutions and keeps both the monotone properties and the high‐order approximation in regions with discontinuities of solutions. The selection criterion has a local character suitable for parallel computation. Application of the proposed technique to the solution of the time‐dependent 2D two‐phase flows through the porous media with the essentially heterogeneous properties is considered, and some numerical results are presented. © 2002 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 18: 44–55, 2002     

Adequacy of finite difference schemes for convection‐diffusion equations

Finite difference schemes for the numerical solution of singularly perturbed convection problems on uniform grids are studied in the limit case where the viscosity and the meshsize approach zero at the same time. The present error estimates are given in terms of order of magnitude in the above limit process and are useful in a priori choosing adequate schemes and meshsizes for boundary‐layer problems and problems with closed characteristics. Published 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 280–295, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10007     

Semi‐implicit spectral collocation methods for reaction‐diffusion equations on annuli

In this article, we develop numerical schemes for solving stiff reaction‐diffusion equations on annuli based on Chebyshev and Fourier spectral spatial discretizations and integrating factor methods for temporal discretizations. Stiffness is resolved by treating the linear diffusion through the use of integrating factors and the nonlinear reaction term implicitly. Root locus curves provide a succinct analysis of the A‐stability of these schemes. By utilizing spectral collocation methods, we avoid the use of potentially expensive transforms between the physical and spectral spaces. Numerical experiments are presented to illustrate the accuracy and efficiency of these schemes. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1113–1129, 2011     

Finite‐difference schemes for transport‐dominated equations using special collocation nodes

We introduce finite‐difference schemes based on a special upwind‐type collocation grid, in order to obtain approximations of the solution of linear transport‐dominated advection‐diffusion problems. The method is well suited when the diffusion parameter is very small compared to the discretization parameter. A theory is developed and many numerical experiments are shown. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

High‐order finite difference schemes for numerical solutions of the generalized Burgers–Huxley equation

In this article, up to tenth‐order finite difference schemes are proposed to solve the generalized Burgers–Huxley equation. The schemes based on high‐order differences are presented using Taylor series expansion. To establish the numerical solutions of the corresponding equation, the high‐order schemes in space and a fourth‐order Runge‐Kutta scheme in time have been combined. Numerical experiments have been conducted to demonstrate the high‐order accuracy of the current algorithms with relatively minimal computational effort. The results showed that use of the present approaches in the simulation is very applicable for the solution of the generalized Burgers–Huxley equation. The current results are also seen to be more accurate than some results given in the literature. The proposed algorithms are seen to be very good alternatives to existing approaches for such physical applications. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1313‐1326, 2011     

High‐order Padé and singly diagonally Runge‐Kutta schemes for linear ODEs,application to wave propagation problems

In this article, we address the problem of constructing high‐order implicit time schemes for wave equations. We consider two classes of one‐step A‐stable schemes adapted to linear Ordinary Differential Equation (ODE). The first class, which is not dissipative is based on the diagonal Padé approximant of exponential function. For this class, the obtained schemes have the same stability function as Gauss Runge‐Kutta (Gauss RK) schemes. They have the advantage to involve the solution of smaller linear systems at each time step compared to Gauss RK. The second class of schemes are constructed such that they require the inversion of a unique linear system several times at each time step like the Singly Diagonally Runge‐Kutta (SDIRK) schemes. While the first class of schemes is constructed for an arbitrary order of accuracy, the second‐class schemes is given up to order 12. The performance assessment we provide shows a very good level of accuracy for both classes of schemes, and the great interest of considering high‐order time schemes that are faster. The diagonal Padé schemes seem to be more accurate and more robust.     

Investigations of nonstandard,Mickens‐type,finite‐difference schemes for singular boundary value problems in cylindrical or spherical coordinates

It is well known that standard finite‐difference schemes for singular boundary value problems involving the Laplacian have difficulty capturing the singular (??(1/r) or ??(log r)) behavior of the solution near the origin (r = 0). New nonstandard finite‐difference schemes that can capture this behavior exactly for certain singular boundary value problems encountered in theoretical aerodynamics are presented here. These schemes are special cases of nonstandard finite differences which have been extensively researched by Professor Ronald E. Mickens of Clark Atlanta University in their most general form. Several examples of these “Mickens‐type” finite differences that illustrate both their accuracy and utility for singular boundary value problems in both cylindrical and spherical co‐ordinates are investigated. The numerical results generated by the Mickens‐type schemes are compared favorably with solutions obtained from standard finite‐difference schemes. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 380–398, 2003.     

High‐resolution relaxation scheme for the two‐dimensional Riemann problems in gas dynamics

We present a new relaxation method for the numerical approximation of the two‐dimensional Riemann problems in gas dynamics. The novel feature of the technique proposed here is that it does not require either a Riemann solver or a characteristics decomposition. The high resolution of the method is achieved by using a third‐order reconstruction for the space discretization and a third‐order TVD Runge‐Kutta scheme for the time integration. Numerical experiments, using several configurations of Riemann problems in gas dynamics, are included to confirm the high resolution of the new relaxation scheme. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Asymptotic preserving numerical schemes for a non‐classical radiation transport model for atmospheric clouds

We present numerical schemes for the P1‐moment and M1‐moment approximations of a non‐classical transport equation modeling radiative transfer in atmospheric clouds. In contrast to classical radiative transfer, the photon path‐length is introduced as an additional variable and serves as pseudo‐time in this model. Because clouds may have optically thick regions, we introduce a diffusive scaling and show that the diffusion limits of the moment models and the original equations agree. Furthermore, we show that the numerical schemes also preserve the diffusion asymptotics as well as the set of admissible and realizable states, both for the explicit and the implicit discretization of the pseudo‐time variable. A source iteration‐like method is proposed, and we observe that it converges slowly in the optical thick case, but a suitable initialization can help to overcome this problem. We validate our method in 1D and present simulation results in the 2D‐case for real cloud data. Copyright © 2013 John Wiley & Sons, Ltd.     

Numerical experiments on a hybrid WENO5 filter for shock‐capturing

In the present paper, a hybrid filter is introduced for high accurate numerical simulation of shock‐containing flows. The fourth‐order compact finite difference scheme is used for the spatial discretization and the third‐order Runge–Kutta scheme is used for the time integration. After each time‐step, the hybrid filter is applied on the results. The filter is composed of a linear sixth‐order filter and the dissipative part of a fifth‐order weighted essentially nonoscillatory scheme (WENO5). The classic WENO5 scheme and the WENO5 scheme with adaptive order (WENO5‐AO) are used to form the hybrid filter. Using a shock‐detecting sensor, the hybrid filter reduces to the linear sixth‐order filter in smooth regions for damping high frequency waves and reduces to the WENO5 filter at shocks in order to eliminate unwanted oscillations produced by the nondissipative spatial discretization method. The filter performance and accuracy of the results are examined through several test cases including the advection, Euler and Navier–Stokes equations. The results are compared with that of a hybrid second‐order filter and also that of the WENO5 and WENO5‐AO schemes.     

Efficient numerical schemes for solving the self‐consistent field equations of flexible–semiflexible diblock copolymers

We present two efficient iterative schemes for solving the self‐consistent field equations of flexible–semiflexible diblock copolymers. One is a semi‐implicit scheme developed by employing asymptotic expansion, and the other is a hybrid scheme combining the robustness of the steepest descent method with the efficiency of the conjugate gradient method. In our position‐one‐dimensional and position‐two‐dimensional numerical experiments, we demonstrate that these schemes are much more efficient than the steepest descent method. Copyright © 2013 John Wiley & Sons, Ltd.     

Stability of global and exponential attractors for a three‐dimensional conserved phase‐field system with memory

We consider a conserved phase‐field system on a tri‐dimensional bounded domain. The heat conduction is characterized by memory effects depending on the past history of the (relative) temperature ?, which is represented through a convolution integral whose relaxation kernel k is a summable and decreasing function. Therefore, the system consists of a linear integrodifferential equation for ?, which is coupled with a viscous Cahn–Hilliard type equation governing the order parameter χ. The latter equation contains a nonmonotone nonlinearity ? and the viscosity effects are taken into account by a term ?αΔ?_tχ, for some α?0. Rescaling the kernel k with a relaxation time ε>0, we formulate a Cauchy–Neumann problem depending on ε and α. Assuming a suitable decay of k, we prove the existence of a family of exponential attractors {?_α,ε} for our problem, whose basin of attraction can be extended to the whole phase–space in the viscous case (i.e. when α>0). Moreover, we prove that the symmetric Hausdorff distance of ?_α,ε from a proper lifting of ?_α,0 tends to 0 in an explicitly controlled way, for any fixed α?0. In addition, the upper semicontinuity of the family of global attractors {??_α,ε} as ε→0 is achieved for any fixed α>0. Copyright © 2009 John Wiley & Sons, Ltd.