We develop mesh conditions for linear finite volume element approximations of anisotropic diffusionconvectionreaction problems to satisfy the discrete maximum principle. We obtain the sufficient conditions to gurantee the both upper and lower bounds of the numerical solution when each angle of arbitrary triangle is O(‖ q‖_∞h+‖ g‖_∞h^2) -acute and h is small enough, where h denotes the mesh size, q and g are coefficients of the convection and reaction terms, respectively. To deal with the convection-dominated problems, we use the upwind triangle technique. For such scheme, the mesh condition can be sharper to O(‖ g‖_∞h^2) -acute. Some numerical examples are presented to demonstrate the theoretical results.

In the construction of cell-centered finite volume schemes for general diffusion equations with discontinuous coefficients on distorted meshes, auxiliary unknowns are usually introduced to resolve discontinuities across cell-edges. In general, it is difficult to express unconditionally auxiliary unknowns as a convex combination of primary unknowns around. As we know, all existing methods have to impose certain restrictive conditions on cell-distortion and coefficient discontinuities, especially when designing discrete schemes satisfying the maximum principle. In this paper, we present a nonlinear convex combination method, in which each vertex unknown is eliminated by a nonlinear convex combination of two cell-centered unknowns, which are respectively the maximum and minimum of those cell-centered unknowns around the vertex. As two application examples we present two cell-centered schemes satisfying the maximum principle based on the nonlinear convex combination. An analysis for the discrete flux shows that our scheme can exactly recover the linear solution. Numerical results are presented to show the accuracy of the resulting schemes and verify the discrete maximum principle.

Some standard numerical methods, such as mimetic finite difference method, finite volume method and mixed finite element method, often fail in solving nonlinear diffusion problems with degenerate diffusion coefficients, since a harmonic average of diffusion coefficients is involved in these methods. To avoid such problem, we present some finite volume element schemes to solve a class of 1D degenerate nonlinear parabolic equations in this paper. Some fully discrete schemes are given by using linear and quadratic finite volume elements in space and a backward difference formulation in time. To deal with unphysical numerical oscillation, we apply two nonnegativity-preserving repair techniques based on a posteriori corrections to finite volume element solutions. One is a local approach, in which any negative energy corresponding to some mesh node is absorbed by the positive values near the current node. The other is a global strategy, in which the total negative energy is redistributed to each positive value in the numerical solution in proportion to its value. In addition, some monotonous finite volume element schemes with lumped-mass strategy are presented. Numerical examples are included to demonstrate the effectiveness and competitive behavior of the proposed methods.

In the numerical simulation for nonlinear diffusion problems with degenerate diffusion coefficients, some classical methods are often invalid since they involve a harmonic mean of diffusion coefficients on the adjacent cells. To avoid such problem, we consider to use linear finite element method to solve a class of 1D degenerate nonlinear parabolic equations. This method can effectively capture the profile of true solution, but at the front it generates some nonphysical numerical oscillations, even brings forth negative values in numerical solution for approximating nonnegative physical quantities. In order to preserve the nonnegativity of true solution, we discuss three repair techniques for finite element solutions based on a posteriori corrections. The first one is a zero-setting method, in which we directly set those negative values to be zero. The second one is a local approach, in which any negative energy associate to some node is absorbed by the positive values around the current node. The third one is a global strategy, in which the total negative energy is redistributed to all positive values with a one-time effort. Numerical examples show that the numerical solution profile is improved remarkably by using the repair techniques for degenerate parabolic equations.

Parallel domain decomposition methods are natural and efficient for solving the implicity schemes of diffusion equations on massive parallel computer systems. A finite volume scheme preserving positivity is essential for getting accurate numerical solutions of diffusion equations and ensuring the numerical solutions with physical meaning. We call their combination as a parallel finite volume scheme preserving positivity, and construct such a scheme for diffusion equation on distorted meshes. The basic procedure of constructing the parallel finite volume scheme is based on the domain decomposition method with the prediction‐correction technique at the interface of subdomains: First, we predict the values on each inner interface of subdomains partitioned by the domain decomposition. Second, we compute the values in each subdomain using a finite volume scheme preserving positivity. Third, we correct the values on each inner interface using the finite volume scheme preserving positivity. The resulting scheme has intrinsic parallelism, and needs only local communication among neighboring processors. Numerical results are presented to show the performance of our schemes, such as accuracy, stability, positivity, and parallel speedup.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2159–2178, 2017

Fully implicit schemes with second‐order time evolutions have been applied to simulate nonlinear diffusion problems precisely for a long time, but there is seldom theoretical study for either their convergence properties or efficient iterations. Here, a second‐order time evolution fully implicit scheme for two‐dimensional nonlinear divergence diffusion problem is analyzed. The unique existence of its solution is given. Two new methods are provided to prove its convergence, including entire inductive hypothesis reasoning and a two‐step reasoning process. Rigorous analysis shows the scheme is stable; its solution has second‐order convergence in both space and time to the exact solution of the problem. The convergence is applied to analyze a Newton iteration accelerating the computation and show its quadratic convergent speed and second‐order accuracy. The reasoning techniques also adapt to first‐order time accuracy schemes, and can be extended to analyze a wide class of nonlinear schemes for nonlinear problems. Numerical tests highlight the theoretical results and demonstrate the high performance of the algorithms. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 121–140, 2016

对于守恒型扩散方程,研究其二阶时间精度非线性全隐有限差分离散格式的性质,证明了其解的存在唯一性.研究了二阶时间精度的Picard-Newton迭代格式,证明了迭代解对原问题真解的二阶时间和空间收敛性,以及对非线性离散解的二次收敛速度,实现了非线性问题的快速求解.本文中方法也适用于一阶时间精度格式的分析,并可推广至对流扩散问题.数值实验验证了二阶时间精度Picard-Newton迭代格式的高精度和高效率.

We construct a new monotone finite volume scheme for diffusion problem on distorted quadrilateral meshes, and the scheme also takes account of the geometric distortion of cells and changes of physical variables on each edge, moreover, a part of the mesh stencil is fixed. The new scheme is proved to be monotone, i.e. it preserves positivity of analytical solutions. Numerical results are presented to demonstrate the numerical performance of our new monotone scheme such as solution positivity-preserving, conservation, accuracy and efficiency on distorted meshes.

本文简要回顾非线性抛物型方程差分方法若干研究工作,包括周毓麟先生在该研究方向取得的部分研究成果,并对近年来相关的部分研究进展进行综述,展望拟开展的研究工作.

A nonlinear finite difference scheme with high accuracy is studied for a class of two-dimensional nonlinear coupled parabolic–hyperbolic system. Rigorous theoretical analysis is made for the stability and convergence properties of the scheme, which shows it is unconditionally stable and convergent with second order rate for both spatial and temporal variables. In the argument of theoretical results, difficulties arising from the nonlinearity and coupling between parabolic and hyperbolic equations are overcome, by an ingenious use of the method of energy estimation and inductive hypothesis reasoning. The reasoning method here differs from those used for linear implicit schemes, and can be widely applied to the studies of stability and convergence for a variety of nonlinear schemes for nonlinear PDE problems. Numerical tests verify the results of the theoretical analysis. Particularly it is shown that the scheme is more accurate and faster than a previous two-level nonlinear scheme with first order temporal accuracy.

For a new nonlinear iterative method named as Picard-Newton (P-N) iterative method for the solution of the time-dependent reaction-diffusion systems, which arise in non-equilibrium radiation diffusion applications, two time step control methods are investigated and a study of temporal accuracy of a first order time integration is presented. The non-equilibrium radiation diffusion problems with flux limiter are considered, which appends pesky complexity and nonlinearity to the diffusion coefficient. Numerical results are presented to demonstrate that compared with Picard method, for a desired accuracy, significant increase in solution efficiency can be obtained by Picard-Newton method with the suitable time step size selection.

A Picard-Newton iteration method is studied to accelerate the numerical solution procedure of a class of two-dimensional nonlinear coupled parabolic-hyperbolic system. The Picard-Newton iteration is designed by adding higher-order terms of small quantity to an existing Picard iteration. The discrete functional analysis and inductive hypothesis reasoning techniques are used to overcome difficulties coming from nonlinearity and coupling, and theoretical analysis is made for the convergence and approximation properties of the iteration scheme. The Picard-Newton iteration has a quadratic convergent ratio, and its solution has second order spatial approximation and first order temporal approximation to the exact solution of the original problem. Numerical tests verify the results of the theoretical analysis, and show the Picard-Newton iteration is more efficient than the Picard iteration. Keywords—nonlinearity, iterative acceleration, coupled parabolichyperbolic system, quadratic convergence, numerical analysis.

A nonlinear iteration method named the Picard–Newton iteration is studied for a two-dimensional nonlinear coupled parabolic–hyperbolic system. It serves as an efficient method to solve a nonlinear discrete scheme with second spatial and temporal accuracy. The nonlinear iteration scheme is constructed with a linearization–discretization approach through discretizing the linearized systems of the original nonlinear partial differential equations. It can be viewed as an improved Picard iteration, and can accelerate convergence over the standard Picard iteration. Moreover, the discretization with second-order accuracy in both spatial and temporal variants is introduced to get the Picard–Newton iteration scheme. By using the energy estimate and inductive hypothesis reasoning, the difficulties arising from the nonlinearity and the coupling of different equation types are overcome. It follows that the rigorous theoretical analysis on the approximation of the solution of the Picard–Newton iteration scheme to the solution of the original continuous problem is obtained, which is different from the traditional error estimate that usually estimates the error between the solution of the nonlinear discrete scheme and the solution of the original problem. Moreover, such approximation is independent of the iteration number. Numerical experiments verify the theoretical result, and show that the Picard–Newton iteration scheme with second-order spatial and temporal accuracy is more accurate and efficient than that of first-order temporal accuracy.

A nonlinear iteration method for solving a class of two-dimensional nonlinear coupled systems of parabolic and hyperbolic equations is studied. A simple iterative finite difference scheme is designed; the calculation complexity is reduced by decoupling the nonlinear system, and the precision is assured by timely evaluation updating. A strict theoretical analysis is carried out as regards the convergence and approximation properties of the iterative scheme, and the related stability and approximation properties of the nonlinear fully implicit finite difference (FIFD) scheme. The iterative algorithm has a linear constringent ratio; its solution gives a second-order spatial approximation and first-order temporal approximation to the real solution. The corresponding nonlinear FIFD scheme is stable and gives the same order of approximation. Numerical tests verify the results of the theoretical analysis. The discrete functional analysis and inductive hypothesis reasoning techniques used in this paper are helpful for overcoming difficulties arising from the nonlinearity and coupling and lead to a related theoretical analysis for nonlinear FI schemes.

Radiative transfer in fluid flow of radiation hydrodynamics is studied. Kinetic laws under radiation condition are investigated. Practical radiation hydrodynamics process is complicated, and numerical simulation is one of primary research means. Splitting methods are often used in numerical simulation, in which fluid motion and radiation are computed separately. We discuss computational problems in radiation diffusion calculations. Diffusion schemes and nonlinear iterative methods on severely distorted meshes are studied. A brief introduction on research progress is given.

A new nonlinear iterative method for nonlinear parabolic equation is developed and ap- plied to a multimaterial nonequilibrium radiation dif- fusion problem on distorted meshes. The new itera- tive method is named by Picard-Newton method (P- N for short). Solution process of the method is as follows. First, by linearizing the time-discretized non- linear partial difierential equation(PDE), we can get an iterative sequence of linear PDE. Then, we de- sign the spatial discretization of the linear PDE, and educe a system of linear algebraic equations. Finally, solve the linear problem by Krylov-subspace meth- ods. The main part of the method is consistent with Picard iterative method, and we can get P-N schemes by adding Newton correction terms to Picard scheme. The e-ciencies of Picard method and Picard-Newton method are compared and the good performance of P-N method is demonstrated.

We construct a monotone finite volume scheme on distorted meshes for multimaterial, nonequilibrium radiation diffusion problems, which are described by the coupled radiation diffusion and material conduction equations. Moreover, we prove theoretically that the scheme is monotone. Numerical results are presented to show that our scheme preserves positivity of solution on various distorted meshes, and the contours of numerical solution obtained by our scheme on distorted meshes accord with that on rectangular meshes. Moreover, numerical tests indicate that our monotone scheme is more computationally efficient than the nine point scheme. These results show that our nonlinear monotone finite volume scheme is a practical and attractive method for solving nonlinear diffusion equations on distorted meshes.

Alternating direction flnite element (ADFE) simulation for moving boundary anisotropic convection difiusion problems is studied. Through the coordinate transformation of the spatial variants, a new domain independent of the time is obtained on which two ADFE algorithms are designed by intro- ducing a small implicit viscous term and approaching the anisotropic difiusion explicitly. Theoretical anal- ysis show that both algorithms have the optimal H 1 and L 2 norm spacial convergency, while their preci- sions for the temporal variant are O(¢t) and O((¢t) 2 ) respectively. Numerical tests are made on three- dimensional model problem to verifled the e-ciency of the algorithms. where ›(t) = fx = (x1;x2;¢¢¢;xd);xi 2 (si;1(t), si;2(t)), i = 1;2;¢¢¢;dg ‰ R d (d ‚ 2 is the dimension of the space), J = (0;T). aij(u) = aij(x;t;u), bi(u) = bi(x;t;u), f(u) = f(x;t;u), u0(x), si;1(t), si;2(t) (i;j = 1;2;¢¢¢;d) are given functions with proper smoothness. Suppose that there exist positive constants s⁄;s ⁄ , a⁄ and a ⁄ such that s⁄ • si;2(t) i si;1(t) • s ⁄ (i = 1;2;¢¢¢;d) and for

A new Lagrangian cell-centered scheme for two-dimensional compressible flows in planar geometry is proposed by Maire et al. The main new feature of the algorithm is that the vertex velocities and the numerical fluxes through the cell interfaces are all evaluated in a coherent manner contrary to standard approaches. In this paper the method introduced by Maire et al. is extended for the equations of Lagrangian gas dynamics in cylindrical symmetry. Two different schemes are proposed, whose difference is that one uses volume weighting and the other area weighting in the discretization of the momentum equation. In the both schemes the conservation of total energy is ensured, and the nodal solver is adopted which has the same formulation as that in Cartesian coordinates. The volume weighting scheme preserves the momentum conservation and the area-weighting scheme preserves spherical symmetry. The numerical examples demonstrate our theoretical considerations and the robustness of the new method.