Monotone Finite Difference Scheme Research Articles

On the Nonstandard Maximum Principle and Its Application for Construction of Monotone Finite-Difference Schemes for Multidimensional Quasilinear Parabolic Equations

On the Nonstandard Maximum Principle and Its Application for Construction of Monotone Finite-Difference Schemes for Multidimensional Quasilinear Parabolic Equations

Numerical Optimal Transport from 1D to 2D Using a Non-local Monge-Ampère Equation

We consider the numerical solution of the optimal transport problem between densities that are supported on sets of unequal dimension. Recent work by McCann and Pass reformulates this problem into a non-local Monge-Ampère type equation. We provide a new level-set framework for interpreting this nonlinear PDE. We also propose a novel discretisation that combines carefully constructed monotone finite difference schemes with a variable-support discrete version of the Dirac delta function. The resulting method is consistent and monotone. These new techniques are described and implemented in the setting of 1D to 2D transport, but they can easily be generalised to higher dimensions. Several challenging computational tests validate the new numerical method.

On the nonstandard maximum principle and its application for construction of monotone finite-difference schemes for multidimensional quasilinear parabolic equations

UDC 517.9 We consider the difference maximum principle with input data of variable sign and its application to the investigation of the monotonicity and convergence of finite-difference schemes (FDSs). Namely, we consider the Dirichlet initial-boundary value problem for multidimensional quasilinear parabolic equation with an unbounded nonlinearity. Unconditionally monotone linearized finite-difference schemes of the second-order of accuracy are constructed on uniform grids. A two-sided estimate for the grid solution, which is completely consistent with similar estimates for the exact solution, is obtained. These estimates are used to prove the convergence of FDSs in the grid L 2 -norm. We also present a study aimed at constructing second-order monotone difference schemes for the parabolic convection-diffusion equation with boundary conditions of the third kind and unlimited nonlinearity without using the initial differential equation on the domain boundaries. The goal is a combination of the assumption of existence and uniqueness of a smooth solution and the regularization principle. In this case, the boundary conditions are directly approximated on a two-point stencil of the second order.

An almost second order uniformly convergent method for a two-parameter singularly perturbed problem with a discontinuous convection coefficient and source term

<p>In this paper, we discuss a higher-order convergent numerical method for a two-parameter singularly perturbed differential equation with a discontinuous convection coefficient and a discontinuous source term. The presence of perturbation parameters generates boundary layers, and the discontinuous terms produce interior layers on both sides of the discontinuity. In order to obtain a higher-order convergent solution, a hybrid monotone finite difference scheme is constructed on a piecewise uniform Shishkin mesh, which is adapted inside the boundary and interior layers. On this mesh (including the point of discontinuity), the present method is almost second-order parameter-uniform convergent. The current scheme is compared with the standard upwind scheme, which is used at the point of discontinuity. The numerical experiments based on the proposed scheme show higher-order (almost second-order) accuracy compared to the standard upwind scheme, which provides almost first-order parameter-uniform convergence.</p>

Three-Dimensional Numerical Transfer Model Using a Monotonized Z-Scheme

The aim of this work is to investigate a three-dimensional second-order monotone finite-difference scheme for the transport equation. The investigation is conducted for model three-dimensional transport equations of an incompressible medium. The properties of the three-dimensional extension of the Z-scheme with nonlinear correction are studied in this work. This study is an extension of the author’s previous works, where a nonlinear correction of one-dimensional transport equations was constructed. The proposed scheme uses “skew differences” containing values from different time layers instead of fluxes for the correction. The monotonicity of the obtained nonlinear scheme is numerically studied for a family of limiter functions for both smooth and non-smooth continuous solutions. The constructed scheme is absolutely stable but loses the monotonicity property when the Courant step is exceeded. The distinctive feature of the proposed finite-difference scheme is the minimalism of its template. The constructed numerical scheme is designed for models of plasma instabilities of various scales in the low-latitude ionospheric plasma of the Earth. One of the real problems that arise in solving such equations is the numerical modeling of strongly non-stationary medium- and small-scale processes in the low-latitude ionosphere of the Earth under conditions of the occurrence of the Rayleigh–Taylor instability and other types of instabilities, leading to the phenomenon of F‑scattering. Due to the fact that transport processes in the ionospheric plasma are controlled by the magnetic field, it is assumed that the plasma is incompressible in the direction perpendicular to the magnetic field.

Hamilton–Jacobi–Bellman–Isaacs equation for rational inattention in the long-run management of river environments under uncertainty

A new stochastic control model for the long-run environmental management of rivers is mathematically and numerically analyzed, focusing on a modern sediment replenishment problem with unique nonsmooth and nonlinear properties. Rational inattention as a novel adaptive strategy to collect information and intervene against the target system is modeled using Erlangization. The system dynamics containing the river discharge following a continuous-state branching with an immigration-type process and the controlled sediment storage dynamics lead to a nonsmooth and nonlocal infinitesimal generator. Modeling uncertainty, which is ubiquitous in certain applications, is considered in a robust control framework in which deviations between the benchmark and distorted models are penalized through relative entropy. The partial integro-differential Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation as an optimality equation is derived, and its uniqueness, existence, and optimality are discussed. A monotone finite difference scheme guaranteeing the boundedness and uniqueness of numerical solutions is proposed to discretize the HJBI equation and is verified based on manufactured solutions. Model applications are also conducted with the parameter values identified from the available data and physical formulae. The computational results suggest that environmental management should be rationally inattentive in a state-dependent and adaptive manner.

Second Order Monotone Finite-Difference Schemes on Non-Uniform Grids for Multi-Dimensional Convection-Diffusion Problem with a Boundary Condition of the Third Kind

Second Order Monotone Finite-Difference Schemes on Non-Uniform Grids for Multi-Dimensional Convection-Diffusion Problem with a Boundary Condition of the Third Kind

The one-phase fractional Stefan problem

We study the existence and properties of solutions and free boundaries of the one-phase Stefan problem with fractional diffusion posed in [Formula: see text]. In terms of the enthalpy [Formula: see text], the evolution equation reads [Formula: see text], while the temperature is defined as [Formula: see text] for some constant [Formula: see text] called the latent heat, and [Formula: see text] stands for the fractional Laplacian with exponent [Formula: see text]. We prove the existence of a continuous and bounded selfsimilar solution of the form [Formula: see text] which exhibits a free boundary at the change-of-phase level [Formula: see text]. This level is located at the line (called the free boundary) [Formula: see text] for some [Formula: see text]. The construction is done in 1D, and its extension to [Formula: see text]-dimensional space is shown. We also provide well-posedness and basic properties of very weak solutions for general bounded data [Formula: see text] in several dimensions. The temperatures [Formula: see text] of these solutions are continuous functions that have finite speed of propagation, with possible free boundaries. We obtain estimates on the growth in time of the support of [Formula: see text] for solutions with compactly supported initial temperatures. Besides, we show the property of conservation of positivity for [Formula: see text] so that the support never recedes. On the contrary, the enthalpy [Formula: see text] has infinite speed of propagation and we obtain precise estimates on the tail. The limits [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are also explored, and we find interesting connections with well-studied diffusion problems. Finally, we propose convergent monotone finite-difference schemes and include numerical experiments aimed at illustrating some of the obtained theoretical results, as well as other interesting phenomena.

A robust numerical method for a two-parameter singularly perturbed time delay parabolic problem

In this article, we consider a class of singularly perturbed two-parameter parabolic partial differential equations with time delay on a rectangular domain. The solution bounds are derived by asymptotic analysis of the problem. We construct a numerical method using a hybrid monotone finite difference scheme on a rectangular mesh which is a product of uniform mesh in time and a layer-adapted Shishkin mesh in space. The error analysis is given for the proposed numerical method using truncation error and barrier function approach, and it is shown to be almost second- and first-order convergent in space and time variables, respectively, independent of both the perturbation parameters. At the end, we present some numerical results in support of the theory.

Convergence & rates for Hamilton–Jacobi equations with Kirchoff junction conditions

We investigate rates of convergence for two approximation schemes of time-independent and time-dependent Hamilton–Jacobi equations with Kirchoff junction conditions. We analyze the vanishing viscosity limit and monotone finite-difference schemes. Following recent work of Lions and Souganidis, we impose no convexity assumptions on the Hamiltonians. For stationary Hamilton–Jacobi equations, we obtain the classical \(\epsilon ^{\frac{1}{2}}\) rate, while we obtain an \(\epsilon ^{\frac{1}{7}}\) rate for approximations of the Cauchy problem. In addition, we present a number of new techniques of independent interest, including a quantified comparison proof for the Cauchy problem and an equivalent definition of the Kirchoff junction condition.

Monotone Finite-Difference Schemes With Second Order Approximation Based on Regularization Approach for the Dirichlet Boundary Problem of the Gamma Equation

We investigate the initial boundary value problem for the Gamma equation transformed from the nonlinear Black-Scholes equation for pricing option to a quasilinear parabolic equation of second derivative. Furthermore, two-side estimates for the exact solution are also provided. By using regularization principle, the unconditionally monotone second order approximation finite-difference scheme on uniform and nonuniform grids is generalized, in that the maximum principle is satisfied without depending on relations of the coefficients and grid parameters. By using the difference maximum principle, we acquired two-side estimates for difference solution for the arbitrary non-sign-constant input data. Finally, we also provide a proof for a priori estimate. It can be confirmed that the two-side estimates for difference solution are completely consistent with the differential problem. Otherwise, the maximal and minimal values of the difference solution is independent from the diffusion and convection coefficients.

Finite-difference method for the Gamma equation on non-uniform grids

We propose a new monotone finite-difference scheme for the second-order local approximation on a nonuniform grid that approximates the Dirichlet initial boundary value problem (IBVP) for the quasi-linear convection-diffusion equation with unbounded nonlinearity, namely, for the Gamma equation obtained by transformation of the nonlinear Black-Scholes equation into a quasilinear parabolic equation. Using the difference maximum principle, a two-sided estimate and an a priori estimate in the C-norm are obtained for the solution of the difference schemes that approximate this equation.

Error estimates for a finite difference scheme associated with Hamilton\u2013Jacobi equations on a junction

This paper is concerned with monotone (time-explicit) finite difference scheme associated with first order Hamilton–Jacobi equations posed on a junction. It extends the scheme introduced by Costeseque et al. (Numer Math 129(3):405–447, 2015) to general junction conditions. On the one hand, we prove the convergence of the numerical solution towards the viscosity solution of the Hamilton–Jacobi equation as the mesh size tends to zero for general junction conditions. On the other hand, we derive some optimal error estimates of in L^{infty }_{text {loc}} for junction conditions of optimal-control type.

Monotone Finite-Difference Schemes of Second-Order Accuracy for Quasilinear Parabolic Equations with Mixed Derivatives

We consider the initial-boundary value problem for quasilinear parabolic equation with mixed derivatives and an unbounded nonlinearity. We construct unconditionally monotone and conservative finite-difference schemes of the second-order accuracy for arbitrary sign alternating coefficients of the equation. For the finite-difference solution, we obtain a two-sided estimate completely consistent with similar estimates for the solution of the differential problem, and also obtain an important a priori estimate in the uniform C-norm. These estimates are used to prove the convergence of finite-difference schemes in the grid L2-norm. All theoretical results are obtained under the assumption that some conditions imposed only on the input data of the differential problem are satisfied.

Improved Accuracy of Monotone Finite Difference Schemes on Point Clouds and Regular Grids

Finite difference schemes are the method of choice for solving nonlinear, degenerate elliptic PDEs, because the Barles--Souganidis convergence framework [G. Barles and P. E. Souganidis, Asymptotic ...

Monotone Finite-Difference Scheme That Preserves the High Accuracy in The Regions of Shock Influence

Monotone Finite-Difference Scheme That Preserves the High Accuracy in The Regions of Shock Influence

Monotone Finite-Difference Scheme Preserving High Accuracy in Regions of Shock Influence

An explicit combined shock-capturing finite-difference scheme is constructed that localizes shock fronts with high accuracy and simultaneously preserves the high order of convergence in all domains where the computed weak solutions are smooth. In this scheme, Rusanov’s explicit nonmonotone scheme of the third order is used as a basis one, while the internal scheme is based on the second-order monotone CABARET. The advantages of the new scheme as compared with the WENO scheme of the fifth order in space and third order in time are demonstrated in test computations.

Mean Field Type Control with Congestion (II): An Augmented Lagrangian Method

This work deals with a numerical method for solving a mean-field type control problem with congestion. It is the continuation of an article by the same authors, in which suitably defined weak solutions of the system of partial differential equations arising from the model were discussed and existence and uniqueness were proved. Here, the focus is put on numerical methods: a monotone finite difference scheme is proposed and shown to have a variational interpretation. Then an Alternating Direction Method of Multipliers for solving the variational problem is addressed. It is based on an augmented Lagrangian. Two kinds of boundary conditions are considered: periodic conditions and more realistic boundary conditions associated to state constrained problems. Various test cases and numerical results are presented.

Monotone Finite Difference Schemes for Anisotropic Diffusion Problems via Nonnegative Directional Splittings

AbstractNonnegative directional splittings of anisotropic diffusion operators in the divergence form are investigated. Conditions are established for nonnegative directional splittings to hold in a neighborhood of an arbitrary interior point. The result is used to construct monotone finite difference schemes for the boundary value problem of anisotropic diffusion operators. It is shown that such a monotone scheme can be constructed if the underlying diffusion matrix is continuous on the closure of the physical domain and symmetric and uniformly positive definite on the domain, the mesh spacing is sufficiently small, and the size of finite difference stencil is sufficiently large. An upper bound for the stencil size is obtained, which is determined completely by the diffusion matrix. Loosely speaking, the more anisotropic the diffusion matrix is, the larger stencil is required. An exception is the situation with a strictly diagonally dominant diffusion matrix where a three-by-three stencil is sufficient for the construction of a monotone finite difference scheme. Numerical examples are presented to illustrate the theoretical findings.

Monotone Finite Difference Schemes for Quasilinear Parabolic Problems with Mixed Boundary Conditions

Abstract In this paper, we consider finite difference methods for two-dimensional quasilinear parabolic problems with mixed Dirichlet–Neumann boundary conditions. Some strong two-side estimates for the difference solution are provided and convergence results in the discrete norm are proved. Numerical examples illustrate the good performance of the proposed numerical approach.