Discrete Maximum Principle Research Articles

An asymptotic-preserving and exactly mass-conservative semi-implicit scheme for weakly compressible flows based on compatible finite elements

We present a novel asymptotic-preserving semi-implicit finite element method for weakly compressible and incompressible flows based on compatible finite element spaces. The momentum is sought in an H(div)-conforming space, ensuring exact pointwise mass conservation at the discrete level. We use an explicit discontinuous Galerkin-based discretization for the nonlinear convective terms, while treating the pressure and viscous terms implicitly, so that the CFL condition depends only on the fluid velocity. To handle shocks and damp spurious oscillations in the compressible regime, we incorporate an a posteriori limiter that employs artificial viscosity and is based on a discrete maximum principle. By using hybridization, the final algorithm requires solving only symmetric positive definite linear systems. As the Mach number approaches zero and the density remains constant, the method naturally converges to an H(div)-based discretization of the incompressible Navier-Stokes equations in the vorticity-velocity-pressure formulation. Several numerical tests validate the proposed method.

Positivity preserving and unconditionally stable numerical scheme for the three-dimensional modified Fisher–Kolmogorov–Petrovsky–Piskunov equation

This paper introduces a numerical approach for the practical solution of the modified Fisher–Kolmogorov–Petrovsky–Piskunov equation that describes population dynamics. The diffusion term and nonlinear term is based on the operator splitting method and interpolation method, respectively. The analytic proof of the discrete maximum principle and positivity preserving for the numerical algorithm is demonstrated. Numerical solution calculated using the proposed method remains stable without blowing up, which implies that the proposed method is unconditionally stable. Numerical studies show that the proposed method is second-order convergence in space and first-order convergence in time. The performance and applicability of the proposed scheme are studied through various computational tests that present the effects of model parameters and evolution dynamics.

A New Second-order Maximum-principle-preserving Finite-volume Method for Flow Problems Involving Discontinuous Coefficients

A new development of Finite Volumes (FV, for short) and its theoretical analysis are the purpose of this work. Recall that FV are known as powerful tools to address equations of conservation laws (mass, energy, momentum,...). Over the last two decades investigators have succeeded in putting in place a mathematical framework for the theoretical analysis of FV. A perfect illustration of this progress is the design and mathematical analysis of Discrete Duality Finite Volumes (DDFV, for short). We propose now a new class of DDFV for 2nd order elliptic equations involving discontinuous diffusion coefficients or nonlinearities. A one-dimensional linear elliptic equation is addressed here for illustrating the ideas behind our numerical strategy. The algebraic structure of the discrete system we have got is different from that of standard DDFV. The main novelty is that the so-called diamond mesh elements are confined in homogeneous zones for flow problems governed by piecewise constant coefficients. This is got from our judicious definition of the primal mesh. The gain is that there is no need to compute homogenized coefficients to be allocated to the so-called diamond cells as required to conventional DDFV. Notice that poor homogenized permeability allocated to diamond elements leads to poor approximations of fluxes across grid-block interfaces. Moreover for 1-D flow problems in a porous medium involving permeability discontinuities (piecewise constant permeability for instance) the proposed FV scheme leads to a symmetric positive-definite discrete system that meets the discrete maximum principle; we have shown its second order convergence under relevant assumptions.

Discontinuous Galerkin schemes for hyperbolic systems in non-conservative variables: Quasi-conservative formulation with subcell finite volume corrections

We present a novel quasi-conservative arbitrary high order accurate ADER (Arbitrary-Derivative) discontinuous Galerkin method allowing to efficiently use a non-conservative form of the considered partial differential system, so that the governing equations can be solved directly in the most physically relevant set of variables. This is particularly interesting for multi-material flows with moving interfaces and steep, large magnitude contact discontinuities, as well as in presence of highly non-linear thermodynamics. However, the non-conservative formulation of course introduces a conservation error which would normally lead to a wrong approximation of shock waves. Hence, from the theoretical point of view, we give a formal definition of the conservation defect of non-conservative schemes and we analyze this defect providing a local quasi-conservation condition, which allows us to prove a modified Lax–Wendroff theorem. Within this formalism, we also reformulate classical results concerning smooth solutions, contact discontinuities and moving interfaces. Then, to deal with shock waves in practice, we exploit the framework of the so-called a posteriori subcell finite volume (FV) limiter, so that, in troubled cells appropriately detected, we can incorporate a local conservation correction. Our corrected FV update entirely removes the local conservation defect, allowing, at least formally, to fit in the hypotheses of the proposed modified Lax–Wendroff theorem. Here, the shock-triggered troubled cells are detected by combining physical admissibility criteria, a discrete maximum principle and a shock sensor inspired by Lagrangian hydrodynamics.To prove the capabilities of our novel approach, first, we show that we are able to recover the same results given by conservative schemes on classical benchmarks for the single-fluid Euler equations. We then conclude the presentation by showing the improved reliability of our scheme on the multi-fluid Euler system on examples like the interaction of a shock with a helium bubble for which we are able to avoid the development of any spurious oscillations.

A nonlocal convection–diffusion model with Gaussian‐type kernels and meshfree discretization

AbstractNonlocal models have demonstrated their indispensability in numerical simulations across a spectrum of critical domains, ranging from analyzing crack and fracture behavior in structural engineering to modeling anomalous diffusion phenomena in materials science and simulating convection processes in heterogeneous environments. In this study, we present a novel framework for constructing nonlocal convection–diffusion models using Gaussian‐type kernels. Our framework uniquely formulates the diffusion term by correlating the constant diffusion coefficient with the variance of the Gaussian kernel. Simultaneously, the convection term is defined by integrating the variable velocity field into the kernel as the expectation of a multivariate Gaussian distribution, facilitating a comprehensive representation of convective transport phenomena. We rigorously establish the well‐posedness of the proposed nonlocal model and derive a maximum principle to ensure its stability and reliability in numerical simulations. Furthermore, we develop a meshfree discretization scheme tailored for numerically simulating our model, designed to uphold both the discrete maximum principle and asymptotic compatibility. Through extensive numerical experiments, we validate the efficacy and versatility of our framework, demonstrating its superior performance compared to existing approaches.

Analysis and numerical methods for nonlocal‐in‐time Allen‐Cahn equation

AbstractIn this paper, we investigate the nonlocal‐in‐time Allen‐Cahn equation (NiTACE), which incorporates a nonlocal operator in time with a finite nonlocal memory. Our objective is to examine the well‐posedness of the NiTACE by establishing the maximal regularity for the nonlocal‐in‐time parabolic equations with fractional power kernels. Furthermore, we derive a uniform energy bound by leveraging the positive definite property of kernel functions. We also develop an energy‐stable time stepping scheme specifically designed for the NiTACE. Additionally, we analyze the discrete maximum principle and energy dissipation law, which hold significant importance for phase field models. To ensure convergence, we verify the asymptotic compatibility of the proposed stable scheme. Lastly, we provide several numerical examples to illustrate the accuracy and effectiveness of our method.

New limiter regions for multidimensional flows

Accurate transport algorithms are crucial for computational fluid dynamics and more accurate and efficient schemes are always in development. One dimensional limiting is commonly employed to suppress nonphysical oscillations. However, the application of such limiters can reduce accuracy. It is important to identify the weakest set of sufficient conditions required on the limiter as to allow the development of successful numerical algorithms.The main goal of this paper is to identify new less restrictive sufficient conditions for flux form in-compressible advection to remain monotonic. We identify additional necessary conditions for incompressible flux form advection to be monotonic, demonstrating that the Spekreijse limiter region is not sufficient for incompressible flux form advection to remain monotonic. Then a convex combination argument is used to derive new sufficient conditions that are less restrictive than the Sweby region for a discrete maximum principle. This allows the introduction of two new more general limiter regions suitable for flux form incompressible advection.

Discrete maximum principle in Hamel’s formalism for nonholonomic optimal control

Discrete maximum principle in Hamel’s formalism for nonholonomic optimal control

A nonlinear finite volume element method preserving the discrete maximum principle for heterogeneous anisotropic diffusion equations

In this paper, we propose a new nonlinear conservative and maximum-principle-preserving finite volume element method for heterogeneous anisotropic diffusion equations on triangular or strictly convex quadrilateral meshes. First, we decompose the numerical flux of the standard finite volume element method on each dual edge into two parts: main part having a two-point-flux structure, and residual part owning a multi-point-flux structure. Then, according to the sign of the residual part and the distribution of the local extremum, the residual part is modified to a two-point-flux structure with a nonlinear term. The resulting nonlinear scheme not only inherits conservation and accuracy of original scheme, but also possesses a maximum-preserving structure without extra restrictions on the meshes and diffusion coefficients. Several numerical experiments verify that our nonlinear scheme has an optimal convergence order and satisfies the discrete maximum principle.

Error Estimates in the Maximum Norm for the Solution of Poisson’s Equation Approximated by the Five-Point Laplacian Using the Discrete Maximum Principle

In this paper, we study error estimates in the maximum norm in the context of solving Poisson’s equation numerically when approximated the Five-Point Laplacian method using the discrete maximum principle. The primary objective is to assess the accuracy of this numerical approach in solving Poisson’s equation and to provide insights into the behavior of error estimates. We focus on the estimates of maximum norm of the discrete functions defined on a grid in a unit square as well as in a square of side s, and estimate errors measured in the maximum norm.

Computational modeling of early-stage breast cancer progression using TPFA method: A numerical investigation

In this paper, a finite volume (TPFA) method is employed to simulate a degenerate breast cancer model that captures the progressive mutations from a normal breast stem cell to a tumor cell. The model incorporates a degenerate parabolic equation to represent the interaction between solid tumor growth and its environment, which involves the release of degradative enzymes governed by a partial differential equation. The discrete maximum principle is verified to be satisfied by the proposed finite volume scheme, and the convergence of the existing discrete solutions to a weak solution is proven. Additionally, the development of breast cancer is demonstrated through a numerical experiment.

A symmetric and coercive finite volume scheme preserving the discrete maximum principle for anisotropic diffusion equations on star-shaped polygonal meshes

A nonlinear vertex-centered finite volume scheme that preserves the discrete maximum principle (DMP) is proposed for anisotropic diffusion equations on polygonal meshes. The new scheme is constructed from a vertex-centered linearity-preserving scheme combined with a nonlinear correction technique. The key ingredient is the introduction of the correction coefficient and the limiter. Particularly, the choice of the limiter is discussed and two new and effective limiters are proposed. Different from most existing nonlinear DMP-preserving schemes, the new nonlinear scheme does not rely on the nonlinear combination of linear flux approximations and requires no interpolation of the auxiliary unknowns. More important is that, it can be proved theoretically that the new scheme is symmetric and coercive on general polygonal meshes with star-shaped cells. Moreover, other theoretical results such as the DMP property and stability of the correction scheme are also presented. Numerical experiments demonstrate the accuracy, efficiency, and DMP property of the new scheme on general polygonal meshes with heterogeneous and anisotropic diffusion tensors.

Sixth-order hybrid finite difference methods for elliptic interface problems with mixed boundary conditions

In this paper, we develop sixth-order hybrid finite difference methods (FDMs) for the elliptic interface problem −∇⋅(a∇u)=f in Ω﹨Γ, where Γ is a smooth interface inside Ω. The variable scalar coefficient a>0 and source f are possibly discontinuous across Γ. The hybrid FDMs utilize a 9-point compact stencil at any interior regular points of the grid and a 13-point stencil at irregular points near Γ. For interior regular points away from Γ, we obtain a sixth-order 9-point compact FDM satisfying the sign and sum conditions for ensuring the M-matrix property. We also derive sixth-order compact (4-point for corners and 6-point for edges) FDMs satisfying the sign and sum conditions for the M-matrix property at any boundary point subject to (mixed) Dirichlet/Neumann/Robin boundary conditions. Thus, for the elliptic problem without interface (i.e., Γ is empty), our compact FDM has the M-matrix property for any mesh size h>0 and consequently, satisfies the discrete maximum principle, which guarantees the theoretical sixth-order convergence. For irregular points near Γ, we propose fifth-order 13-point FDMs, whose stencil coefficients can be effectively calculated by recursively solving several small linear systems. Theoretically, the proposed high order FDMs use high order (partial) derivatives of the coefficient a, the source term f, the interface curve Γ, the two jump functions along Γ, and the functions on ∂Ω. Numerically, we always use function values to approximate all required high order (partial) derivatives in our hybrid FDMs without losing accuracy. Our proposed FDMs are independent of the choice representing Γ and are also applicable if the jump conditions on Γ only depend on the geometry (e.g., curvature) of the curve Γ. Our numerical experiments confirm the sixth-order convergence in the l∞ norm of the proposed hybrid FDMs for the elliptic interface problem.

Fitted finite element methods for singularly perturbed elliptic problems of convection-diffusion type

Fitted finite element methods are constructed for a singularly perturbed convection-diffusion problem in two space dimensions. Exponential splines as basis functions are combined with Shishkin meshes to obtain a stable parameter-uniform numerical method. These schemes satisfy a discrete maximum principle. If the diffusivity parameter is bounded below by some fixed positive constant, the numerical approximations converge, in L∞, at a rate of second order. Moreover, the numerical approximations converge at a rate of first order for all values of the singular perturbation parameter.

Elliptic operators with discontinuous coefficients in meshfree GFDM

AbstractIn phase change simulations, material properties such as density, viscosity, or thermal conductivity may exhibit jump discontinuities, possibly of several orders of magnitude. These jump discontinuities represent interfaces between the phases, and they emerge naturally during the simulation; thus, their exact location is generally unknown a priori. Our goal is to simulate phase change processes with a meshfree generalized finite difference method in a monolithic model without distinguishing between the different phases. There, the material properties mentioned above appear as coefficients inside elliptic operators in divergence form and the jumps must be treated adequately by the numerical method. We present a numerical method for discretizing elliptic operators with discontinuous coefficients without the need for a domain decomposition or tracking of interfaces. Our method facilitates the construction of diagonally dominant diffusion operators that lead to M‐matrices for the discrete Poisson's equation, and thus, satisfy the discrete maximum principle. We demonstrate the applicability of the new method for the case of smooth diffusivity and discontinuous diffusivity. We show that the method is first‐order accurate for discontinuous diffusion problems and provides second‐order and fourth‐order convergence for continuous diffusion coefficients.

Fitted exact difference method for solutions of a singularly perturbed time delay parabolic PDE

A novel numerical method has been proposed for solving singularly perturbed parabolic time delay PDE by using a fitted mesh exact finite difference method. The traditional numerical methods on uniform mesh fail to accurately approximate the solution of the considered PDE, due to the presence of a boundary layer. Therefore, developing a numerical scheme that can effectively handle the boundary layer behaviour is of great interest. The proposed scheme utilizes Crank–Nicolson method for temporal discretization and fitted mesh exact finite difference method for spatial discretization. The scheme satisfies the discrete maximum principle, stability bound, and uniform convergence with the boundary layer resolving property. Numerical test examples have been conducted to validate the scheme for different values of the perturbation parameter and mesh numbers. The results show that the proposed scheme outperforms existing methods in the literature in terms of accuracy. The proposed approach holds significant potential for solving similar PDEs that exhibit boundary layer behaviour.

Fitted computational method for singularly perturbed convection-diffusion equation with time delay

A uniformly convergent numerical scheme is proposed to solve a singularly perturbed convection-diffusion problem with a large time delay. The diffusion term of the problem is multiplied by a perturbation parameter, ε. For a small ε, the problem exhibits a boundary layer, which makes it challenging to solve it analytically or using standard numerical methods. As a result, the backward Euler scheme is applied in the temporal direction. Non-symmetric finite difference schemes are applied for approximating the first-order derivative terms, and a higher-order finite difference method is applied for approximating the second-order derivative term. Furthermore, an exponential fitting factor is computed and induced in the difference scheme to handle the effect of the small parameter. Using the discrete maximum principle, the stability of the scheme is examined and analyzed. The developed scheme is parameter-uniform with a linear order of convergence in both space and time. To examine the accuracy of the method, two model examples are considered. Further, the boundary layer behavior of the solutions is given graphically.

A nonlinear scheme preserving maximum principle for heterogeneous anisotropic diffusion equation

We present a new nonlinear method in order to preserve the maximum principle for the diffusion problem with heterogeneous anisotropic coefficient. It is well-known that for diffusion problems with general discontinuous coefficients, the existing cell-centered finite volume schemes on general meshes cannot unconditionally satisfy the discrete maximum principle (DMP), i.e., there are always certain severe restrictions on the mesh-cell geometry or locations of discontinuity in order to preserve DMP. To deal with the issue we propose a nonlinear method of modifying discrete flux of linear second-order schemes into a new conservative flux. Then the resulting scheme keeps conservation and accuracy, and has the structure preserving maximum principle without imposing those severe restrictions. Specifically, we will start with the diamond scheme whose flux is linear and has second-order accuracy, but has no the structure preserving maximum principle. Then the nonlinear correction method is applied to get a scheme satisfying the discrete maximum principle unconditionally. We give some theoretical analysis including the coercivity property and a prior estimation. Numerical results show that the accuracy of our new scheme is superior to the existing scheme preserving maximum principle, and in some cases the accuracy can even be improved by one order of magnitude.

A finite volume scheme preserving the invariant region property for a class of semilinear parabolic equations on distorted meshes

AbstractIn this article, we present a finite volume scheme preserving invariant‐region‐property (IRP) for a class of semilinear parabolic equations with anisotropic diffusion coefficient on distorted meshes. The diffusion term is discretized by the finite volume scheme preserving the discrete maximum principle, and the time derivative is discretized by the backward Euler scheme. For the nonlinear system, a specially designed iteration is proposed to preserve the IRP. The IRPs are proved for both, the finite volume scheme and the nonlinear iteration. Numerical examples are presented to verify the accuracy and IRP of our scheme.

A Reinforcement Learning Based Slope Limiter for Second‐Order Finite Volume Schemes

AbstractHyperbolic equations admit discontinuities in the solution and thus adequate and physically sound numerical schemes are necessary for their discretization. Second‐order finite volume schemes are a popular choice for the discretization of hyperbolic problems due to their simplicity. Despite the numerous advantages of higher‐order schemes in smooth regions, they fail at strong discontinuities. Crucial for the accurate and stable simulation of flow problems with discontinuities is the adequate and reliable limiting of the reconstructed slopes. Numerous limiters have been developed to handle this task. However, they are too dissipative in smooth regions or require empirical parameters which are globally defined and test case specific. Therefore, this paper aims to develop a new slope limiter based on deep learning and reinforcement learning techniques. For this, the proposed limiter is based on several admissibility constraints: positivity of the solution and a relaxed discrete maximum principle. This approach enables a slope limiter which is independent of a manually specified global parameter while providing an optimal slope with respect to the defined admissibility constraints. The new limiter is applied to several well‐known shock tube problems, which illustrates its broad applicability and the potential of reinforcement learning in numerics.