In this paper, we propose a high-order fully implicit spatio-temporal discrete scheme, referred to as the BDF-DDG scheme, for solving nonlinear reaction-diffusion systems containing self-and cross-diffusion on arbitrary polygonal meshes. The scheme combines the Direct Discontinuous Galerkin (DDG) method for spatial discretization with the Backward Differentiation Formula (BDF) for temporal discretization, thereby achieving high-order accuracy in both space and time. The nonlinear reaction terms are addressed through the application of the Picard-Newton iterative method, which offers both flexibility and efficiency in execution. The theoretical foundation of the scheme is established under relaxed assumption for the diffusion matrix K, which follows a UP-decomposition framework. In this framework, U is a symmetric positive definite matrix, and P is a diagonal matrix with non-negative diagonal elements. This assumption not only includes the case where K is symmetric positive definite, but also includes the situation where K itself is not symmetric but satisfies the UP decomposition. Under this foundation, the numerical solution in L-2-norm can achieve optimal (p+ 1)th order accuracy through polynomial approximation of degree p in space and kth order accuracy by using the BDF-k (1 <= k <= 5) discrete scheme in time. Numerical experiments further demonstrate that the BDF-DDG scheme exhibits both temporal and spatial high accuracy through computations of various practical reaction-diffusion problems. These results highlight the robustness and reliability of the numerical scheme, establishing it as a powerful and efficient simulation tool for complex biological and chemical systems.

In this article, we aim to develop a high order direct discontinuous Galerkin (DDG) method solving elliptic interface problem on arbitrary polygon fitted meshes. Elliptic interface problem with the homogeneous or non-homogeneous interface conditions can be solved in the uniform discrete DDG formulation. Numerical analysis results show that high order DDG method for polygonal elliptic interface problem on arbitrary polygon fitted meshes can reach the optimal kth order convergence in H-1 norm and the optimal (k + 1)th order convergence in L-2 norm. By refining meshes nearby the curved interface, errors coming from polygonal approximating interface can be reduced. A sequence of numerical experiments are carried out to verify the optimal convergence of DDG method with high order P-2, P-3 and P-4 approximations to deal with several different interface situations. It means that DDG method is capable of handing interface problems with complicated geometries meshes and interface situations.

This paper presents the stability of a two-layer coupled discretization fully implicit finite element scheme as well as the uniqueness of its solution. The scheme has been proposed for solving multi-dimensional anisotropic diffusion equations with nonlinear capacity term, and the existence and convergence of its solution have been proved in [Fang et al., J. Comput. Appl. Math. 438 (2024) 115512]. However, the basic theoretical analysis is incomplete, for example, the stability and uniqueness have not been solved yet, which are very important for the application of numerical methods in engineering. In this paper, we further develop the discrete functional analysis techniques to establish a framework with relatively comprehensive theoretical results. Wherein by introducing Ritz projection, rewriting the error equations into equivalent forms, and choosing appropriate test functions, we propose a new inductive argument to overcome the difficulties arising from the coupled nonlinear discretization of the diffusion operator and capacity term. Consequently, on the basis of the existence and convergence properties, we prove for the first time that the nonlinear finite element method is stable, thereby its solution is unique. Numerical examples show that the scheme is stable and has no numerical oscillations compared with the classical Crank–Nicolson scheme.

A new nonlinear finite element method is studied to solve multi-dimensional anisotropic nonlinear diffusion equation with nonlinear capacity term, especially to secure a vivid simulation in the case of problems with transient physical quantities. In the scheme design, fully implicit two-layer coupled discretization is applied to assure high time accuracy, and finite element discretization is applied to assure high space accuracy. By introducing Ritz projection and new inductive reasoning methods, we develop the discrete functional analysis technique from that for finite difference schemes, and establish a new general analysis framework for nonlinear finite element schemes. Consequently, we overcome the difficulties arising from the strong coupling of the nonlinear finite element discretizations for the capacity term and diffusion operator, and prove the existence and boundedness of the nonlinear finite element solution, as well as its second-order time and optimal order space convergence. Numerical experiments and comparisons confirm the theoretical analysis results and demonstrate its high performance.

Some fundamental properties are analyzed for a fully implicit finite difference (FIFD) solution of conservative strong nonlinear diffusion problem. The scheme is constructed by combining a second-order backward difference temporal discretization and a central finite difference spatial discretization, and therefore highly nonlinear. Theoretical analysis is carried out under a coercive condition according with the diffusion feature of the strong nonlinear diffusion model. Benefiting from the boundedness estimates of the FIFD solution itself and its first- and second-order spatial difference quotients, some novel argument techniques are developed to overcome the difficulties coming from the nonlinear approximation for the strong nonlinear conservative diffusion operator. Consequently, it is proved rigorously that the FIFD scheme is unconditionally stable, its solution is unique and convergent to the exact solution of the original problem with second-order space-time accuracy. Numerical examples are provided to confirm its advantages on precision and efficiency over its first-order time accurate counterpoint.

A nonlinear fully implicit finite difference scheme with second-order time evolution for nonlinear diffusion problem is studied. The scheme is constructed with two-layer coupled discretization (TLCD) at each time step. It does not stir numerical oscillation, while permits large time step length, and produces more accurate numerical solutions than the other two well-known second-order time evolution nonlinear schemes, the Crank-Nicolson (CN) scheme and the backward difference formula second-order (BDF2) scheme. By developing a new reasoning technique, we overcome the difficulties caused by the coupled nonlinear discrete diffusion operators at different time layers, and prove rigorously the TLCD scheme is uniquely solvable, unconditionally stable, and has second-order convergence in both space and time. Numerical tests verify the theoretical results, and illustrate its superiority over the CN and BDF2 schemes.

A fully implicit finite difference scheme with second-order time evolution for strong nonlinear diffusion equation of divergence type is studied. Theoretical analysis is established under a coercive condition reflecting the diffusion characteristics of the strong nonlinear equation. Some new reasoning techniques are developed to overcome the difficulties caused by the strong nonlinearity of the divergence diffusion operator. The existence of the fully implicit finite difference solution is proved by constructing an appropriate mapping in the fixed point argument and bounding the solution as well as its first- and second-order spatial difference quotients tactfully.

A fully implicit finite difference (FIFD) scheme with second-order space-time accuracy is studied for a nonlinear diffusion equation with general capacity term. A new reasoning procedure is introduced to overcome difficulties caused by the nonlinearity of the capacity term and the diffusion operator in the theoretical analysis. The existence of the FIFD solution is investigated at first which plays an important role in the analysis. It is established by choosing a new test function to bound the solution and its temporal and spatial difference quotients in suitable norms in the fixed point arguments, which is different from the traditional way. Based on these bounds, other fundamental properties of the scheme are rigorously analyzed consequently. It shows that the scheme is uniquely solvable, unconditionally stable, and convergent with second-order space-time accuracy inL(infinity)(L-2)andL(infinity)(H-1)norms. The theoretical analysis adapts to both one- and multidimensional problems, and can be extended to schemes with first-order time accuracy. Numerical tests are provided to verify the theoretical results and highlight the high accuracy of the second-order space-time accurate scheme. The reasoning techniques can be extended to a broad family of discrete schemes for nonlinear problems with capacity terms.

This paper discusses accelerating iterative methods for solving the fully implicit (FI) scheme of equilibrium radiation diffusion problem. Together with the FI Picard factorization (PF) iteration method, three new nonlinear iterative methods, namely, the FI Picard-Newton factorization (PNF), FI Picard-Newton (PN) and derivative free Picard-Newton factorization (DFPNF) iteration methods are studied, in which the resulting linear equations can preserve the parabolic feature of the original PDE. By using the induction reasoning technique to deal with the strong nonlinearity of the problem, rigorous theoretical analysis is performed on the fundamental properties of the four iteration methods. It shows that they all have first-order time and second-order space convergence, and moreover, can preserve the positivity of solutions. It is also proved that the iterative sequences of the PF iteration method and the three Newton-type iteration methods converge to the solution of the FI scheme with a linear and a quadratic speed respectively. Numerical tests are presented to confirm the theoretical results and highlight the high performance of these Newton acceleration methods.

For a two-layer finite difference scheme with second-order time accuracy of nonlinear diffusion equations, we present three iterative solving algorithms, including Picard, Picard-Newton and derivative-free Picard-Newton iterations. The main purpose of this paper is to solve the two-layer scheme efficiently and accurately, and give strict theoretical proofs of the convergence and efficiency of the three iterative methods. For the three iterative methods, two strategies are adopted to offer iterative initial evaluations, one is to use the value at the previous time step with first-order accuracy, the other is an extrapolation with second-order accuracy. With an induction reasoning technique, we prove that the solutions of the three iterative methods all converge to the exact solution of the diffusion problem with second-order accuracy both in space and time after two nonlinear iteration steps, even if the first-order initial value is used. It is also proved that the solutions of Picard iteration converge linearly to the solution of the discrete scheme, while Picard-Newton and derivative-free Picard-Newton iterative solutions converge with a quadratic speed. Moreover, no difference occurs in convergent speed with the two different initial values. Finally, some numerical tests are presented to verify our theoretical results, which show that compared with Picard iteration, Picard-Newton and derivative-free Picard-Newton iterations are more efficient for solving nonlinear problems.

The carbuncle phenomenon is a numerical instability that affects the numerical capturing of shock waves when low-dissipative upwind scheme is used. This paper investigates shock instabilities of the HLLE-type methods for the Euler equations under the strong shock interaction, where the HLLE-type methods include the HLLE, HLLC, HLLEM, HLLCM and HLLEC Riemann solvers with specific wavespeed estimates. Based on a matrix stability analysis for two dimensional steady shocks, a new factor to influence carbuncle phenomenon is pointed out and the choice of the signal velocity plays an important role. A numerical flux function with wave velocity estimates which can crisply resolve shocks seems to be vulnerable to the shock anomalies even if the numerical fluxes to be regarded to be free from the carbuncle phenomenon. A suggestion to the choice of the wave speed is proposed when calculating strong shock wave problems.

Purpose This paper aims to provide a well-behaved nonlinear scheme and accelerating iteration for the nonlinear convection diffusion equation with fundamental properties illustrated. Design/methodology/approach A nonlinear finite difference scheme is studied with fully implicit (FI) discretization used to acquire accurate simulation. A Picard–Newton (PN) iteration with a quadratic convergent ratio is designed to realize fast solution. Theoretical analysis is performed using the discrete function analysis technique. By adopting a novel induction hypothesis reasoning technique, the L∞ (H1) convergence of the scheme is proved despite the difficulty because of the combination of conservative diffusion and convection operator. Other properties are established consequently. Furthermore, the algorithm is extended from first-order temporal accuracy to second-order temporal accuracy. Findings Theoretical analysis shows that each of the two FI schemes is stable, its solution exists uniquely and has second-order spatial and first/second-order temporal accuracy. The corresponding PN iteration has the same order of accuracy and quadratic convergent speed. Numerical tests verify the conclusions and demonstrate the high accuracy and efficiency of the algorithms. Remarkable acceleration is gained. Practical implications The numerical method provides theoretical and technical support to accelerate resolving convection diffusion, non-equilibrium radiation diffusion and radiation transport problems. Originality/value The FI schemes and iterations for the convection diffusion problem are proposed with their properties rigorously analyzed. The induction hypothesis reasoning method here differs with those for linearization schemes and is applicable to other nonlinear problems.

We study the asymptotic-preserving fully discrete schemes for nonequilibrium radiation diffusion problem in spherical and cylindrical symmetric geometry.The research is based on two-temperature models with Larsen's flux-limited diffusion operators.Finite volume spatially discrete schemes are developed to circumvent the singularity at the origin and the polar axis and assure local conservation.Asymmetric second order accurate spatial approximation is utilized instead of the traditional first order one for boundary flux-limiters to consummate the schemes with higher order global consistency errors.The harmonic average approach in spherical geometry is analyzed, and its second order accuracy is demonstrated.By formal analysis, we prove these schemes and their corresponding fully discrete schemes with implicitly balanced and linearly implicit time evolutions have first order asymptoticpreserving properties.By designing associated manufactured solutions and reference solutions, we verify the desired performance of the fully discrete schemes with numerical tests, which illustrates quantitatively they are first order asymptotic-preserving and basically second order accurate, hence competent for simulations of both equilibrium and non-equilibrium radiation diffusion problems.

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

Motivated by providing well-behaved fully discrete schemes in practice, this paper extends the asymptotic analysis on time integration methods for non-equilibrium radiation diffusion in [2] to space discretizations. Therein studies were carried out on a two-temperature model with Larsen's flux-limited diffusion operator, both the implicitly balanced (IB) and linearly implicit (LI) methods were shown asymptotic-preserving. In this paper, we focus on asymptotic analysis for space discrete schemes in dimensions one and two. First, in construction of the schemes, in contrast to traditional first-order approximations, asymmetric second-order accurate spatial approximations are devised for flux-limiters on boundary, and discrete schemes with second-order accuracy on global spatial domain are acquired consequently. Then by employing formal asymptotic analysis, the first-order asymptotic-preserving property for these schemes and furthermore for the fully discrete schemes is shown. Finally, with the help of manufactured solutions, numerical tests are performed, which demonstrate quantitatively the fully discrete schemes with IB time evolution indeed have the accuracy and asymptotic convergence as theory predicts, hence are well qualified for both non-equilibrium and equilibrium radiation diffusion.

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.

Weighted interior penalty discontinuous Galerkin method is developed to solve the two-dimensional non-equilibrium radiation diffusion equation on unstructured mesh. There are three weights including the arithmetic, the harmonic, and the geometric weight in the weighted discontinuous Galerkin scheme. For the time discretization, we treat the nonlinear diffusion coefficients explicitly, and apply the semi-implicit integration factor method to the nonlinear ordinary differential equations arising from discontinuous Galerkin spatial discretization. The semi-implicit integration factor method can not only avoid severe timestep limits, but also takes advantage of the local property of DG methods by which small sized nonlinear algebraic systems are solved element by element with the exact Newton iteration method. Numerical results are presented to demonstrate the validity of discontinuous Galerkin method for high nonlinear and tightly coupled radiation diffusion equation.

We present a Lax-Wendroff discontinuous Galerkin (LWDG) method combining with adaptive mesh refinement (AMR) to solve three-dimensional hyperbolic conservation laws. Compared with Runge-Kutta discontinuous finite element method (RKDG) the method has higher efficiency. We give an effective adaptive strategie. Equidistribution strategy is easily implemented on nonconforming tetrahedral mesh. Error indicator is introduced to solve three-dimensional Euler equations. Numerical experiments demonstrate that the method has satisfied numerical efficiency.

This paper discusses the weighted discontinuous Galerkin method for elliptic problems with jump coefficients.For the two-dimensional linear elliptic equation with discontinuous coefficient,we propose a new weighted symmetric interior penalty method.The convergence analysis is presented based on the coercivity and continuity of bilinear form.Numerical examples demonstrate the validity of DG method for elliptic problems with strongly discontinuous coefficients.

In this paper, a new discontinuous Galerkin method is developed for the parabolic equation with jump coefficients satisfying the continuous flow condition. Theoretical analysis shows that this method is L-2 stable. When the finite element space consists of interpolative polynomials of degrees k, the convergent rate of the semi-discrete discontinuous Galerkin scheme has an order of O(h(k)). Numerical examples for both 1-dimensional and 2-dimensional problems demonstrate the validity of the new method.