High-order accurate monotone difference schemes for solving gasdynamic problems by Godunov’s method with antidiffusion

An approach to the construction of high-order accurate monotone difference schemes for solving gasdynamic problems by Godunov’s method with antidiffusion is proposed. Godunov’s theorem on monotone schemes is used to construct a new antidiffusion flux limiter in high-order accurate difference schemes as applied to linear advection equations with constant coefficients. The efficiency of the approach is demonstrated by solving linear advection equations with constant coefficients and one-dimensional gasdynamic equations.     

A family of explicit parallel Runge-Kutta-Nyström methods

A procedure for the construction of high-order explicit parallel Runge-Kutta-Nyström (RKN) methods for solving second-order nonstiff initial value problems (IVPs) is analyzed. The analysis reveals that starting the procedure with a reference symmetric RKN method it is possible to construct high-order RKN schemes which can be implemented in parallel on a small number of processors. These schemes are defined by means of a convex combination of k disjoint s_i-stage explicit RKN methods which are constructed by connecting s_i steps of a reference explicit symmetric method. Based on the reference second-order Störmer-Verlet methods we derive a family of high-order explicit parallel schemes which can be implemented in variable-step codes without additional cost. The numerical experiments carried out show that the new parallel schemes are more efficient than some sequential and parallel codes proposed in the scientific literature for solving second-order nonstiff IVPs.     

HIGH-ORDER NONLINEAR GALERKIN METHODS

The nonlinear Galerkin methods are numerical schemes well adapted to the long-term integration of nonlinear evolution partial differential equations. In this paper, a class of high-order nonlinear Galerkin methods are provided. Moreover, convergence results with high-order spectral accuracy are derived for the schemes introduced.     

Two-stage complex Rosenbrock schemes for stiff systems

New two-stage Rosenbrock schemes with complex coefficients are proposed for stiff systems of differential equations. The schemes are fourth-order accurate and satisfy enhanced stability requirements. A one-parameter family of L1-stable schemes with coefficients explicitly calculated by formulas involving only fractions and radicals is constructed. A single L2-stable scheme is found in this family. The coefficients of the fourth-order accurate L4-stable scheme previously obtained by P.D Shirkov are refined. Several fourth-order schemes are constructed that are high-order accurate for linear problems and possess the limiting order of L-decay. The schemes proposed are proved to converge. A symbolic computation algorithm is developed that constructs order conditions for multistage Rosenbrock schemes with complex coefficients. This algorithm is used to design the schemes proposed and to obtain fifth-order accurate conditions.     

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).     

High-order computational methods for option valuation under multifactor models

Many of the different numerical techniques in the partial differential equations framework for solving option pricing problems have employed only standard second-order discretization schemes. A higher-order discretization has the advantage of producing low size matrix systems for computing sufficiently accurate option prices and this paper proposes new computational schemes yielding high-order convergence rates for the solution of multi-factor option problems. These new schemes employ Galerkin finite element discretizations with quadratic basis functions for the approximation of the spatial derivatives in the pricing equations for stochastic volatility and two-asset option problems and time integration of the resulting semi-discrete systems requires the computation of a single matrix exponential. The computations indicate that this combination of high-order finite elements and exponential time integration leads to efficient algorithms for multi-factor problems. Highly accurate European prices are obtained with relatively coarse meshes and high-order convergence rates are also observed for options with the American early exercise feature. Various numerical examples are provided for illustrating the accuracy of the option prices for Heston’s and Bates stochastic volatility models and for two-asset problems under Merton’s jump-diffusion model.     

High-order accurate implicit running schemes

An approach to the construction of high-order accurate implicit predictor-corrector schemes is proposed. The accuracy is improved by choosing a special time integration step for computing numerical fluxes through cell interfaces by using an unconditionally stable implicit scheme. For smooth solutions of advection equations with constant coefficients, the scheme is second-order accurate. Implicit difference schemes for multidimensional advection equations are constructed on the basis of Godunov’s method with splitting over spatial variables as applied to the computation of “large” values at an intermediate layer. The numerical solutions obtained for advection equations and the radiative transfer equations in a vacuum are compared with their exact solutions. The comparison results confirm that the approach is efficient and that the accuracy of the implicit predictor-corrector schemes is improved.     

A TVD Type Wavelet-Galerkin Method for Hamilton-Jacobi Equations

In this paper,we use Daubechies scaling functions as test functions for the Galerkin method,and discuss Wavelet-Galerkin solutions for the Hamilton-Jacobi equations.It can be proved that the schemesare TVD schemes.Numerical tests indicate that the schemes are suitable for the Hamilton-Jacobi equations.Furthermore,they have high-order accuracy in smooth regions and good resolution of singularities.     

Multioperator representation of composite compact schemes

The multioperator approach is used to obtain high-order accurate compact differences. These differences are developed to describe convective terms of differential equations, as well as mixed derivatives, source terms, and the coefficients of metric derivatives of coordinate transformations. The same principles are used to obtain high-order compact differences for representing diffusion terms. These differences underlie multioperator composite compact schemes, which are used to compute the flow past an airfoil by integrating the nonstationary Navier-Stokes equations supplemented with the equations of a turbulent viscosity model.     

High-order ADI method for stream-function vorticity equations

Many recent works have demonstrated the efficiency of high-order compact (HOC) difference schemes on the stream-function and vorticity formulation of 2-D incompressible Navier-Stokes equations. HOC discretizations induce cross spatial derivatives which are treated explicitly in most ADI schemes. Recently, Karaa and Zhang proposed a fourth-order ADImethod for solving convection-diffusion problems efficiently. In this work, we extend this method to the solution of incompressible Navier-Stokes equations. The driven flow in a square cavity is used as a model problem and numerical results are compared with other results. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Monotonicity criteria for difference schemes designed for hyperbolic equations

Previously formulated monotonicity criteria for explicit two-level difference schemes designed for hyperbolic equations (S.K. Godunov’s, A. Harten’s (TVD schemes), characteristic criteria) are extended to multileveled, including implicit, stencils. The characteristic monotonicity criterion is used to develop a universal algorithm for constructing high-order accurate nonlinear monotone schemes (for an arbitrary form of the desired solution) based on their analysis in the space of grid functions. Several new fourth-to-third-order accurate monotone difference schemes on a compact three-level stencil and nonexpanding (three-point) stencils are proposed for an extended system, which ensures their monotonicity for both the desired function and its derivatives. The difference schemes are tested using the characteristic monotonicity criterion and are extended to systems of hyperbolic equations.     

求解时间分布阶扩散方程的两个高阶有限差分格式

基于复化Simpson公式和复化两点Gauss-Legendre公式，构造了两个求解时间分布阶扩散方程的高阶有限差分格式.不同于以往文献中提出的时间一阶或二阶格式，这两种格式在时间方向都具有三阶精度，而在分布阶和空间方向可达到四阶精度.数值结果表明，两种算法都是稳定且收敛的，从而是有效的.两种格式的收敛速率也通过数值实验进行了验证，并且通过和文献中的算法对比可以得出其更为高效,     

Hybrid schemes with high-order multioperators for computing discontinuous solutions

Results are presented concerning high-order multioperator schemes and their monotonized versions as applied to the computation of discontinuous solutions. Two types of hybrid schemes are considered. Solutions of several test problems, including those with extremely strong discontinuities, are presented. An example of solving the Navier-Stokes equations at low supersonic Mach numbers by applying multioperator schemes without monotonization is given.     

Fourth-order compact schemes with adaptive time step for monodomain reaction–diffusion equations

Multigrid applied to fourth-order compact schemes for monodomain reaction–diffusion equations in two dimensions has been developed. The scheme accounts for the anisotropy of the medium, allows for any cellular activation model to be used, and incorporates an adaptive time step algorithm. Numerical simulations show up to a 40% reduction in computational time for complex cellular models as compared to second-order schemes for the same solution error. These results point to high-order schemes as valid alternatives for the efficient solution of the cardiac electrophysiology problem when complex cellular activation models are used.     

Adaptive high-order splitting schemes for large-scale differential Riccati equations

We consider high-order splitting schemes for large-scale differential Riccati equations. Such equations arise in many different areas and are especially important within the field of optimal control. In the large-scale case, it is critical to employ structural properties of the matrix-valued solution, or the computational cost and storage requirements become infeasible. Our main contribution is therefore to formulate these high-order splitting schemes in an efficient way by utilizing a low-rank factorization. Previous results indicated that this was impossible for methods of order higher than 2, but our new approach overcomes these difficulties. In addition, we demonstrate that the proposed methods contain natural embedded error estimates. These may be used, e.g., for time step adaptivity, and our numerical experiments in this direction show promising results.     

Application of the Chebyshev pseudospectral method to van der Waals fluids

In this paper, we consider a class of van der Waals flows with non-convex flux functions. In these flows, nonclassical under-compressive shock waves can develop. Such waves, which are characterized by kinetic functions, violate classical entropy conditions. We propose to use a Chebyshev pseudospectral method for solving the governing equations. A comparison of the results of this method with very high-order (up to 10th-order accurate) finite difference schemes is presented, which shows that the proposed method leads to a lower level of numerical oscillations than other high-order finite difference schemes and also does not exhibit fast-traveling packages of short waves which are usually observed in high-order finite difference methods. The proposed method can thus successfully capture various complex regimes of waves and phase transitions in both elliptic and hyperbolic regimes.     

Minimal dissipation hybrid bicompact schemes for hyperbolic equations

New monotonicity-preserving hybrid schemes are proposed for multidimensional hyperbolic equations. They are convex combinations of high-order accurate central bicompact schemes and upwind schemes of first-order accuracy in time and space. The weighting coefficients in these combinations depend on the local difference between the solutions produced by the high- and low-order accurate schemes at the current space-time point. The bicompact schemes are third-order accurate in time, while having the fourth order of accuracy and the first difference order in space. At every time level, they can be solved by marching in each spatial variable without using spatial splitting. The upwind schemes have minimal dissipation among all monotone schemes constructed on a minimum space-time stencil. The hybrid schemes constructed has been successfully tested as applied to a number of two-dimensional gas dynamics benchmark problems.     

Numerical algorithms for diffusion-reaction problems with non-classical conditions

Parabolic equations with nonlocal boundary conditions have been given considerable attention in recent years. In this paper new high-order algorithms for the linear diffusion-reaction problem are derived. The convergence of the new schemes is studied and numerical examples are given to show the efficiency of the new methods to solve linear and nonlinear diffusion-reaction equations with these non classical conditions.     

Formalism of two potentials for the numerical solution of Maxwell’s equations

A new formulation of Maxwell’s equations based on the introduction of two vector and two scalar potentials is proposed. As a result, the electromagnetic field equations are written as a hyperbolic system that contains, in contrast to the original Maxwell system, only evolution equations and does not involve equations in the form of differential constraints. This makes the new equations especially convenient for the numerical simulation of electromagnetic processes. Specifically, they can be solved by applying powerful modern shock-capturing methods based on the approximation of spatial derivatives by upwind differences. The cases of an electromagnetic field in a vacuum and an inhomogeneous material are considered. Examples are given in which electromagnetic wave propagation is simulated by solving the formulated system of equations with the help of modern high-order accurate schemes.     

Parallel Implicit-Explicit General Linear Methods

High-order discretizations of partial differential equations(PDEs)necessitate high-order time integration schemes capable of handling both stiff and nonstiff op...