RKDG methods with WENO type limiters and conservative interfacial procedure for one-dimensional compressible multi-medium flow simulations

In this paper, we continue on studying the Runge-Kutta discontinuous Galerkin (RKDG) methods to solve compressible multi-medium flow with conservative treatment of the moving material interface. Comparing with the paper by J. Qiu, T.G. Liu and B.C. Khoo [J. Comput. Phys. 222 (2007) 353-373], we adopt the HLLC flux instead of Lax-Friedrichs numerical flux, the finite volume weighted essentially nonoscillatory (WENO) and Hermite WENO (HWENO) reconstructions as limiter instead of TVB limiter for RKDG. The HLLC flux is based on the approximate Riemann solver with little numerical viscosity and can resolve the contact discontinuity and shear wave very well. For limiter procedure, first we use the KXRCF indicator to identify the troubled cell, then apply WENO or HWENO method to reconstruct the polynomial in the troubled cell, while maintaining the cell average. This limiter procedure is more accurate and less problem dependent than the TVB limiter. Numerical results in one dimension for multi-medium flows such as gas-gas and gas-water are provided to illustrate the capability of these procedures.     

A Survey of High Order Schemes for the Shallow Water Equations

In this paper, we survey our recent work on designing high order positivity-preserving well-balanced finite difference and finite volume WENO (weighted essentially non-oscillatory) schemes, and discontinuous Galerkin finite element schemes for solving the shallow water equations with a non-flat bottom topography. These schemes are genuinely high order accurate in smooth regions for general solutions, are essentially non-oscillatory for general solutions with discontinuities, and at the same time they preserve exactly the water at rest or the more general moving water steady state solutions. A simple positivity-preserving limiter, valid under suitable CFL condition, has been introduced in one dimension and reformulated to two dimensions with triangular meshes, and we prove that the resulting schemes guarantee the positivity of the water depth.     

The method of lines for the computation of incompressible viscous flows

A common approach to finding numerical solutions of the time-dependent incompressible Navier-Stokes equations is considered within the method of lines framework [1]. For the determination of viscous incompressible flows the stream-function formulation for the fourth-order equation [2, 3], an artificial compressibility method [4], and a modified velocity correction method [5] are designed. Some improved and extended results of numerical simulations obtained by the author in the previous works are presented. Test calculations have been done for various flows inside square, triangular and semicircular cavities with one moving wall, the backward-facing step, double bent channels and for the flow around an aerofoil at large angle of attack. An alternative and practical methodology for resolving the Navier-Stokes equations in arbitrarily complex geometries using Cartesian meshes is proposed. Some of complex geometrical configurations can be decomposed into a set of subdomains. The simplest approach for specifying boundary conditions near curved or irregular boundaries is to transfer all the variables from the boundaries to the nearest grid knots. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Simulation of elastic wave propagation in geological media: Intercomparison of three numerical methods

For wave propagation in heterogeneous media, we compare numerical results produced by grid-characteristic methods on structured rectangular and unstructured triangular meshes and by a discontinuous Galerkin method on unstructured triangular meshes as applied to the linear system of elasticity equations in the context of direct seismic exploration with an anticlinal trap model. It is shown that the resulting synthetic seismograms are in reasonable quantitative agreement. The grid-characteristic method on structured meshes requires more nodes for approximating curved boundaries, but it has a higher computation speed, which makes it preferable for the given class of problems.     

A positive spatial advection scheme on unstructured meshes for tracer transport

Modelling tracer transport (leading to a single linear advection–diffusion equation) for realistic data provides a challenging task with respect to the robustness of the underlying numerical procedures. In this paper, we contribute at this point by formulating a positive spatial advection scheme for unstructured triangular meshes. The proof of positivity is presented in detail, using an elementary classification. It is shown that with a careful reconstruction procedure and a moderate demand towards the grid a positive advection scheme is obtained. Next, a brief discussion is given on how we implement this scheme in combination with an implicit time-stepping procedure. As a numerical example, we discuss tracer transport in a strongly heterogeneous porous medium.     

Note on the cubic decreasing region of the Chebyshev method

In this paper we reduce the two-dimensional cubic decreasing region considered in Hernandez and Salanova (2000) [1], [2] into one-dimensional region or interval for the Chebyshev method. It means that we find a simple sufficient condition for the semilocal convergence of the method.     

A unified framework of multi-objective cost functions for partitioning unstructured finite element meshes

Good performance of parallel finite element computations on unstructured meshes requires high-quality mesh partitioning. Such a decomposition task is normally done by a graph-based partitioning approach. However, the main shortcoming of graph partitioning algorithms is that minimizing the so-called edge cut is not entirely the same as minimizing the communication overhead. This paper thus proposes a unified framework of multi-objective cost functions, which take into account several factors that are not captured by the graph-based partitioning approach. Freely adjustable weighting parameters in the framework also promote a flexible treatment of different optimization objectives. A greedy-style post-improvement procedure is designed to use these cost functions to improve the quality of subdomain meshes arising from the graph-based partitioning approach. Both serial and parallel implementation of the post-improvement procedure have been done. Numerical experiments show that communication overhead can indeed be reduced by this improvement procedure, thereby increasing the performance of parallel finite element computations.     

Binary weighted essentially non-oscillatory (BWENO) approximation

Weighted essentially non-oscillatory (WENO) schemes have been mainly used for solving hyperbolic partial differential equations (PDEs). Such schemes are capable of high order approximation in smooth regions and non-oscillatory sharp resolution of discontinuities. The base of the WENO schemes is a non-oscillatory WENO approximation procedure, which is not necessarily related to PDEs. The typical WENO procedures are WENO interpolation and WENO reconstruction. The WENO algorithm has gained much popularity but the basic idea of approximation did not change much over the years. In this paper, we first briefly review the idea of WENO interpolation and propose a modification of the basic algorithm. New approximation should improve basic characteristics of the approximation and provide a more flexible framework for future applications. New WENO procedure involves a binary tree weighted construction that is based on key ideas of WENO algorithm and we refer to it as the binary weighted essentially non-oscillatory (BWENO) approximation. New algorithm comes in a rational and a polynomial version. Furthermore, we describe the WENO reconstruction procedure, which is usually involved in the numerical schemes for hyperbolic PDEs, and propose the new reconstruction procedure based on the described BWENO interpolation. The obtained numerical results show that the newly proposed procedures perform very well on the considered test examples.     

Planar mesh refinement cannot be both local and regular

We show that two desirable properties for planar mesh refinement techniques are incompatible. Mesh refinement is a common technique for adaptive error control in generating unstructured planar triangular meshes for piecewise polynomial representations of data. Local refinements are modifications of the mesh that involve a fixed maximum amount of computation, independent of the number of triangles in the mesh. Regular meshes are meshes for which every interior vertex has degree 6. At least for some simple model meshing problems, optimal meshes are known to be regular, hence it would be desirable to have a refinement technique that, if applied to a regular mesh, produced a larger regular mesh. We call such a technique a regular refinement. In this paper, we prove that no refinement technique can be both local and regular. Our results also have implications for non-local refinement techniques such as Delaunay insertion or Rivara's refinement. Received August 1, 1996 / Revised version received February 28, 1997     

An asymptotic preserving scheme with the maximum principle for the M1 model on distorded meshes

In this Note, we show that a recent scheme introduced by Buet et al. (2011) [5] for the nonlinear two moments $M_1$ model of linear transport and which captures correctly the diffusion limit on distorded meshes (AP scheme) also possesses the maximum principle. The main idea of the design of this scheme is to rewrite the model as a gas dynamics model and to use an Eulerian scheme, derived from a Lagrange + remap scheme. To obtain the AP property we use the multidimensional extension, developed by Buet et al. (2012) [6], of the Jin and Levermore (1996) procedure [9] for the hyperbolic heat equation. We will show that this scheme is entropic which ensures the maximum principle of the $M_1$ model. More we present some numerical results, on distorted quadrangular and triangular meshes which show that the scheme is second order in the diffusive regime.     

Convergence of a staggered Lax-Friedrichs scheme for nonlinear conservation laws on unstructured two-dimensional grids

Summary. Based on Nessyahu and Tadmor's nonoscillatory central difference schemes for one-dimensional hyperbolic conservation laws [16], for higher dimensions several finite volume extensions and numerical results on structured and unstructured grids have been presented. The experiments show the wide applicability of these multidimensional schemes. The theoretical arguments which support this are some maximum-principles and a convergence proof in the scalar linear case. A general proof of convergence, as obtained for the original one-dimensional NT-schemes, does not exist for any of the extensions to multidimensional nonlinear problems. For the finite volume extension on two-dimensional unstructured grids introduced by Arminjon and Viallon [3,4] we present a proof of convergence for the first order scheme in case of a nonlinear scalar hyperbolic conservation law. Received April 8, 2000 / Published online December 19, 2000     

Symmetrical weighted essentially non‐oscillatory‐flux limiter schemes for Hamilton–Jacobi equations

In this paper, we propose a new scheme that combines weighted essentially non‐oscillatory (WENO) procedures together with monotone upwind schemes to approximate the viscosity solution of the Hamilton–Jacobi equations. In one‐dimensional (1D) case, first, we obtain an optimum polynomial on a four‐point stencil. This optimum polynomial is third‐order accurate in regions of smoothness. Next, we modify a second‐order ENO polynomial by choosing an additional point inside the stencil in order to obtain the highest accuracy when combined with the Harten–Osher reconstruction‐evolution method limiter. Finally, the optimum polynomial is considered as a symmetric and convex combination of three polynomials with ideal weights. Following the methodology of the classic WENO procedure, then, we calculate the non‐oscillatory weights with the ideal weights. Numerical experiments in 1D and 2D are performed to compare the capability of the hybrid scheme to WENO schemes. Copyright © 2015 John Wiley & Sons, Ltd.     

Grid-characteristic method on unstructured tetrahedral meshes

The goal of this paper is to develop a grid-characteristic method intended for high-performance computer systems and implemented on unstructured tetrahedral hierarchical meshes with the use of a multiple time step and high-order interpolation, including interpolation with a limiter, piecewise parabolic interpolation, and monotone interpolation. The method is designed for simulating complex three-dimensional dynamical processes in heterogeneous media. It involves accurately stated contact conditions and produces physically correct solutions of problems in seismology and seismic exploration. Hierarchical meshes make it possible to take into account numerous inhomogeneous inclusions (cracks, cavities, etc.) and to solve problems in a real-life formulation. The grid-characteristic method enables the use of a multiple time step. As a result, the computation time is considerably reduced and the efficiency of the method is raised. The method is parallelized on a computer cluster with an optimal use of system resources.     

A hybridizable discontinuous triangular spectral element method on unstructured meshes and its hp-error estimates

Numerical Algorithms - In this paper, a hybridizable discontinuous triangular spectral element method (HDTSEM) using tensorial nodal basis functions on unstructured meshes is proposed and analyzed....     

A new method based on SOM network to generate coarse meshes for overlapping unstructured multigrid algorithm

A new method to generate coarse meshes for overlapping unstructured multigrid algorithm based on self-organizing map (SOM) neural network is presented in this paper. The application of SOM neural network can overcome some limitations of conventional methods and which is designed to pursuit the best structure relation between fine and coarse unstructured meshes with the object to ensure robust convergence for overlapping unstructured multigrid algorithm. Besides, this method can automate the generation of unstructured meshes and is suitable for both two and three dimensions conditions.     

On the positivity of linear weights in WENO approximations

High order accurate weighted essentially non-oscillatory （WENO） schemes have been used extensively in numerical solutions of hyperbolic partial differential equations and other convection dominated problems. However the WENO procedure can not be applied directly to obtain a stable scheme when negative linear weights are present. In this paper, we first briefly review the WENO framework and the role of linear weights, and then present a detailed study on the positivity of linear weights in a few typical WENO procedures, including WENO interpolation, WENO reconstruction and WENO approximation to first and second derivatives, and WENO integration. Explicit formulae for the linear weights are also given for these WENO procedures. The results of this paper should be useful for future design of WENO schemes involving interpolation, reconstruction, approximation to first and second derivatives, and integration procedures.     

An algorithm for coarsening unstructured meshes

Summary. We develop and analyze a procedure for creating a hierarchical basis of continuous piecewise linear polynomials on an arbitrary, unstructured, nonuniform triangular mesh. Using these hierarchical basis functions, we are able to define and analyze corresponding iterative methods for solving the linear systems arising from finite element discretizations of elliptic partial differential equations. We show that such iterative methods perform as well as those developed for the usual case of structured, locally refined meshes. In particular, we show that the generalized condition numbers for such iterative methods are of order , where is the number of hierarchical basis levels. Received December 5, 1994     

High order accurate semi-implicit WENO schemes for hyperbolic balance laws

In this paper we propose a family of well-balanced semi-implicit numerical schemes for hyperbolic conservation and balance laws. The basic idea of the proposed schemes lies in the combination of the finite volume WENO discretization with Roe’s solver and the strong stability preserving (SSP) time integration methods, which ensure the stability properties of the considered schemes [S. Gottlieb, C.-W. Shu, E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001) 89-112]. While standard WENO schemes typically use explicit time integration methods, in this paper we are combining WENO spatial discretization with optimal SSP singly diagonally implicit (SDIRK) methods developed in [L. Ferracina, M.N. Spijker, Strong stability of singly diagonally implicit Runge-Kutta methods, Appl. Numer. Math. 58 (2008) 1675-1686]. In this way the implicit WENO numerical schemes are obtained. In order to reduce the computational effort, the implicit part of the numerical scheme is linearized in time by taking into account the complete WENO reconstruction procedure. With the proposed linearization the new semi-implicit finite volume WENO schemes are designed.A detailed numerical investigation of the proposed numerical schemes is presented in the paper. More precisely, schemes are tested on one-dimensional linear scalar equation and on non-linear conservation law systems. Furthermore, well-balanced semi-implicit WENO schemes for balance laws with geometrical source terms are defined. Such schemes are then applied to the open channel flow equations. We prove that the defined numerical schemes maintain steady state solution of still water. The application of the new schemes to different open channel flow examples is shown.     

Efficient implementation of WENO schemes to nonuniform meshes

Most of the standard papers about the WENO schemes consider their implementation to uniform meshes only. In that case the WENO reconstruction is performed efficiently by using the algebraic expressions for evaluating the reconstruction values and the smoothness indicators from cell averages. The coefficients appearing in these expressions are constant, dependent just on the scheme order, not on the mesh size or the reconstruction function values, and can be found, for example, in Jiang and Shu (J Comp Phys 126:202–228, 1996). In problems where the geometrical properties must be taken into account or the solution has localized fine scale structure that must be resolved, it is computationally efficient to do local grid refinement. Therefore, it is also desirable to have numerical schemes, which can be applied to nonuniform meshes. Finite volume WENO schemes extend naturally to nonuniform meshes although the reconstruction becomes quite complicated, depending on the complexity of the grid structure. In this paper we propose an efficient implementation of finite volume WENO schemes to nonuniform meshes. In order to save the computational cost in the nonuniform case, we suggest the way for precomputing the coefficients and linear weights for different orders of WENO schemes. Furthermore, for the smoothness indicators that are defined in an integral form we present the corresponding algebraic expressions in which the coefficients obtained as a linear combination of divided differences arise. In order to validate the new implementation, resulting schemes are applied in different test examples.