Finite volume schemes on triangles coupled with multiresolution analysis

A multiresolution procedure is used to reduce the costs of flux evaluations in a finite volume scheme. A two-dimensional hyperbolic conservation law is solved on the finest grid among a hierarchy of nested grids. The mean values of the solution on triangles of a given grid are estimated from the coarser level using an original reconstruction algorithm. The size of the differences between the mean values and their reconstruction is a local regularity criterium and dictates the choice of the flux computation method. Numerical experiments with computing time comparisons are presented.     

Fully adaptive multiresolution finite volume schemes for conservation laws

The use of multiresolution decompositions in the context of finite volume schemes for conservation laws was first proposed by A. Harten for the purpose of accelerating the evaluation of numerical fluxes through an adaptive computation. In this approach the solution is still represented at each time step on the finest grid, resulting in an inherent limitation of the potential gain in memory space and computational time. The present paper is concerned with the development and the numerical analysis of fully adaptive multiresolution schemes, in which the solution is represented and computed in a dynamically evolved adaptive grid. A crucial problem is then the accurate computation of the flux without the full knowledge of fine grid cell averages. Several solutions to this problem are proposed, analyzed, and compared in terms of accuracy and complexity.

     

Multiresolution schemes for strongly degenerate parabolic equations in one space dimension

An adaptive finite volume method for one‐dimensional strongly degenerate parabolic equations is presented. Using an explicit conservative numerical scheme with a third‐order Runge‐Kutta method for the time discretization, a third‐order ENO interpolation for the convective term, and adding a conservative discretization for the diffusive term, we apply the multiresolution method combining two fundamental concepts: the switch between central interpolation or exact computing of numerical flux and a thresholded wavelet transform applied to cell averages of the solution to control the switch. Applications to mathematical models of sedimentation‐consolidation processes and traffic flow with driver reaction, which involve different types of boundary conditions, illustrate the computational efficiency of the new method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

A multiresolution space‐time adaptive scheme for the bidomain model in electrocardiology

The bidomain model of electrical activity of myocardial tissue consists of a possibly degenerate parabolic PDE coupled with an elliptic PDE for the transmembrane and extracellular potentials, respectively. This system of two scalar PDEs is supplemented by a time‐dependent ODE modeling the evolution of the gating variable. In the simpler subcase of the monodomain model, the elliptic PDE reduces to an algebraic equation. Since typical solutions of the bidomain and monodomain models exhibit wavefronts with steep gradients, we propose a finite volume scheme enriched by a fully adaptive multiresolution method, whose basic purpose is to concentrate computational effort on zones of strong variation of the solution. Time adaptivity is achieved by two alternative devices, namely locally varying time stepping and a Runge‐Kutta‐Fehlberg‐type adaptive time integration. A series of numerical examples demonstrates that these methods are efficient and sufficiently accurate to simulate the electrical activity in myocardial tissue with affordable effort. In addition, the optimal choice of the threshold for discarding nonsignificant information in the multiresolution representation of the solution is addressed, and the numerical efficiency and accuracy of the method is measured in terms of CPU time speed‐up, memory compression, and errors in different norms. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

A Well-Balanced HLL Scheme for Hyperbolic Conservation Systems With Source Terms北大核心CSCD

针对含源项的双曲守恒方程给出了一种新的有限体积格式.经典的有限体积格式不能正确地模拟对流通量项和外力之间的平衡所产生的动力学问题.为解决这个问题,仿照经典的HLL近似Riemann求解器设计思路设计了含源项的近似Riemann求解器.针对含重力源项的一维流体Euler方程和理想磁流体方程,通过对通量计算格式的修正得到了保平衡HLL格式（WB-HLL）,并给出了保平衡的证明.针对一维Euler方程和理想磁流体给出了两个算例,比较了传统HLL格式和提出的WB-HLL格式的计算精度.计算结果表明,WB-HLL格式精度更高,收敛更快.     

An adaptive multiresolution method on dyadic grids: Application to transport equations

We propose a modified adaptive multiresolution scheme for solving d$d$-dimensional hyperbolic conservation laws which is based on cell-average discretization in dyadic grids. Adaptivity is obtained by interrupting the refinement at the locations where appropriate scale (wavelet) coefficients are sufficiently small. One important aspect of such a multiresolution representation is that we can use the same binary tree data structure for domains of any dimension. The tree structure allows us to succinctly represent the data and efficiently navigate through it. Dyadic grids also provide a more gradual refinement as compared with the traditional quad-trees (2D) or oct-trees (3D) that are commonly used for multiresolution analysis. We show some examples of adaptive binary tree representations, with significant savings in data storage when compared to quad-tree based schemes. As a test problem, we also consider this modified adaptive multiresolution method, using a dynamic binary tree data structure, applied to a transport equation in 2D domain, based on a second-order finite volume discretization.     

A finite volume method for scalar conservation laws with stochastic time–space dependent flux functions

We propose a new finite volume method for scalar conservation laws with stochastic time–space dependent flux functions. The stochastic effects appear in the flux function and can be interpreted as a random manner to localize the discontinuity in the time–space dependent flux function. The location of the interface between the fluxes can be obtained by solving a system of stochastic differential equations for the velocity fluctuation and displacement variable. In this paper we develop a modified Rusanov method for the reconstruction of numerical fluxes in the finite volume discretization. To solve the system of stochastic differential equations for the interface we apply a second-order Runge–Kutta scheme. Numerical results are presented for stochastic problems in traffic flow and two-phase flow applications. It is found that the proposed finite volume method offers a robust and accurate approach for solving scalar conservation laws with stochastic time–space dependent flux functions.     

A multiresolution finite volume scheme for two-dimensional hyperbolic conservation laws

In this paper, high-resolution finite volume schemes are combined with an adaptive mesh technique inspired by multiresolution analysis to improve the computational efficiency for two-dimensional hyperbolic conservation laws. The method is conservative. Moreover, it is stable which is proven numerically in this paper. The computational grid is dynamically adapted so that higher spatial resolution is automatically allocated to regions where strong gradients are observed. Using this proposed scheme, we compute several two-dimensional model problems and a compressive rate ranging from about 5–10 is observed in all simulations.     

A complete flux scheme for the stationary advection-diffusion-reaction equation

Expressions for the numerical flux of a conservation law of advection-diffusion-reaction type are derived from a local solution of the entire conservation law, including the source term. The resulting complete flux scheme is given for one-dimensional (in Cartesian and spherical coordinates) and two-dimensional model equations. Combined with a finite volume method, the numerical scheme is second order accurate, uniformly in the Peclet numbers. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

基于通量ENO形式的多分辨算法

1．数值方法在这一章中,我们将以一维情况为例来简要介绍多分辨算法的思想,详细情况请参阅［9,1,2,11,12］．首先考虑一维双曲方程的初值问题这里u（x,t）是含q个分量的向量,我们假定Jacobi矩阵o八on有q个实特征值al三rZ…三Zq,相应的右特征向量,1,…,lq可展成＊q空间．为了使问题简单起见,在这里我们假定。0和u均为周期为2的周期函数．因此在所有以下计算中除非特别说明我们将应用周期性边界条件·令丐一、仇,0三j三认为区间你2］的等分点,矾22”‘,n。为整数．现在将方程（1）写为如下的全离散形式,由方程（3）可以知道…     

Multiresolution algorithms for the numerical solution of hyperbolic conservation laws

Given any scheme in conservation form and an appropriate uniform grid for the numerical solution of the initial value problem for one-dimensional hyperbolic conservation laws we describe a multiresolution algorithm that approximates this numerical solution to a prescribed tolerance in an efficient manner. To do so we consider the grid-averages of the numerical solution for a hierarchy of nested diadic grids in which the given grid is the finest, and introduce an equivalent multiresolution representation. The multiresolution representation of the numerical solution consists of its grid-averages for the coarsest grid and the set of errors in predicting the grid-averages of each level of resolution in this hierarchy from those of the next coarser one. Once the numerical solution is resolved to our satisfaction in a certain locality of some grid, then the prediction errors there are small for this particular grid and all finer ones; this enables us to compress data by setting to zero small components of the representation which fall below a prescribed tolerance. Therefore instead of computing the time-evolution of the numerical solution on the given grid we compute the time-evolution of its compressed multiresolution representation. Algorithmically this amounts to computing the numerical fluxes of the given scheme at the points of the given grid by a hierarchical algorithm which starts with the computation of these numerical fluxes at the points of the coarsest grid and then proceeds through diadic refinements to the given grid. At each step of refinement we add the values of the numerical flux at the center of the coarser cells. The information in the multiresolution representation of the numerical solution is used to determine whether the solution is locally well-resolved. When this is the case we replace the costly exact value of the numerical flux with an accurate enough approximate value which is obtained by an inexpensive interpolation from the coarser grid. The computational efficiency of this multiresolution algorithm is proportional to the rate of data compression (for a prescribed level of tolerance) that can be achieved for the numerical solution of the given scheme.     

Well-balanced unstaggered central schemes for one and two-dimensional shallow water equation systems

We propose a new well-balanced unstaggered central finite volume scheme for hyperbolic balance laws with geometrical source terms. In particular we construct a new one and two-dimensional finite volume method for the numerical solution of shallow water equations on flat/variable bottom topographies. The proposed scheme evolves a non-oscillatory numerical solution on a single grid, avoids the time consuming process of solving Riemann problems arising at the cell interfaces, and is second-order accurate both in space and time. Furthermore, the numerical scheme follows a well-balanced discretization that first discretizes the geometrical source term according to the discretization of the flux terms, and then mimics the surface gradient method and discretizes the water height according to the discretization of the water level. The resulting scheme exactly satisfies the C-property at the discrete level. The proposed scheme is then applied and classical one and two-dimensional shallow water equation problems with flat or variable bottom topographies are successfully solved. The obtained numerical results are in good agreement with corresponding ones appearing in the recent literature, thus confirming the potential and efficiency of the proposed method.     

Monotone combined edge finite volume–finite element scheme for Anisotropic Keller–Segel model

In this article, a new numerical scheme for a degenerate Keller–Segel model with heterogeneous anisotropic tensors is treated. It is well‐known that standard finite volume scheme not permit to handle anisotropic diffusion without any restrictions on meshes. Therefore, a combined finite volume‐nonconforming finite element scheme is introduced, developed, and studied. The unknowns of this scheme are the values at the center of cell edges. Convergence of the approximate solution to the continuous solution is proved only supposing the shape regularity condition for the primal mesh. This scheme ensures the validity of the discrete maximum principle under the classical condition that all transmissibilities coefficients are positive. Therefore, a nonlinear technique is presented, as a correction of the diffusive flux, to provide a monotone scheme for general tensors. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1030–1065, 2014     

A Finite Volume Scheme for Savage-Hutter Equations on Unstructured Grids

A Godunov-type finite volume scheme on unstructured grids is proposed to numerically solve the Savage-Hutter equations in curvilinear coordinate. We show the direct observation that the model isn't a Galilean invariant system. At the cell boundary, the modified Harten-Lax-van Leer (HLL) approximate Riemann solver is adopted to calculate the numerical flux. The modified HLL flux is not troubled by the lack of Galilean invariance of the model and it is helpful to handle discontinuities at free interface. Rigidly the system is not always a hyperbolic system due to the dependence of flux on the velocity gradient. Even so, our numerical results still show quite good agreements with reference solutions. The simulations for granular avalanche flows with shock waves indicate that the scheme is applicable.     

Locally divergence-preserving schemes of higher order

Violation of the divergence constraint on the magnetic flux density in magnetohydrodynamical (MHD) simulations leads to stability problems. It is therefore of great importance to numerically respect this intrinsic constraint. Since the divergence preservation is a local phenomenon inherent in the MHD-system it is appealing to mimic this property numerically by a locally divergence-preserving scheme. A common numerical technique for simulation of the MHD-system of conservation laws is the finite volume (FV) method. In [SISC 26 2005 pp. 1166] a local procedure to redistribute the numerical fluxes in a FV-scheme so that a discrete divergence operator vanishes was presented. This procedure stabilizes the base scheme and respects the accuracy to the second order level. The present note describes a development of the above procedure that complies with the finite volume framework, preserves a fourth order discrete divergence operator locally and retains the accuracy of a generic semi-discrete finite volume scheme up to fourth order. The redistribution of the numerical magnetic field fluxes is formulated in a standard conservative setting, making it trivial to implement the divergence-preserving modification in an existing FV-scheme; see [JCP 227 2008 pp.3405] for the details. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Estimation d'erreur pour une loi de conservation scalaire multidimensionelle approchée par un schéma implicite de volumes finis

In this Note we present an error estimate for the approximate solution of the nonlinear hyperbolic equation u_t + div(ƒ(u(x, t))v(x)) = 0 by an implicit finite volume scheme. We show that the error is of order √k +√h, where h and k are respectively the sizes of the space and the time steps parameters. A generalisation of this result to an arbitrary consistent monotone numerical flux is presented in [4]. The convergence of this scheme is possible with k/h (CFL condition) going to infinity.     

On the convergence of high resolution methods with multiple time scales for hyperbolic conservation laws

A class of finite volume methods based on standard high resolution schemes, but which allows spatially varying time steps, is described and analyzed. A maximum principle and the TVD property are verified for general advective flux, extending the previous theoretical work on local time stepping methods. Moreover, an entropy condition is verified which, with sufficient limiting, guarantees convergence to the entropy solution for convex flux.

     

多边形网格上扩散方程新的单调格式

 本文在星形多边形网格上, 构造了扩散方程新的单调有限体积格式.该格式与现有的基于非线性两点流的单调格式的主要区别是, 在网格边的法向流离散模板中包含当前边上的点, 在推导离散法向流的表达式时采用了定义于当前边上的辅助未知量, 这样既可适应网格几何大变形, 同时又兼顾了当前网格边上物理量的变化. 在光滑解情形证明了离散法向流的相容性.对于具有强各向异性、非均匀张量扩散系数的扩散方程, 证明了新格式是单调的, 即格式可以保持解析解的正性. 数值结果表明在扭曲网格上, 所构造的格式是局部守恒和保正的, 对光滑解有高于一阶的精度, 并且, 针对非平衡辐射限流扩散问题, 数值结果验证了新格式在计算效率和守恒精度上优于九点格式.     

An implicit finite volume method for the solution of 3D low Mach number viscous flows using a local preconditioning technique

This paper presents a cell-centered high order finite volume scheme for the solution of the three-dimensional (3D) Navier–Stokes equations with low Mach number. The system of non-linear equations is solved by means of a fully implicit pseudo-transient scheme. Each pseudo-time step is solved by a Newton-GMRes procedure. A local preconditioning technique is used to scale the speed of sound and to improve the system condition number for low Mach number and low cell Reynolds number. This preconditioning is applied to the AUSM+up flux vector splitting function. The method is tested on 2D and 3D low Mach number laminar flows.