Numerical Methods for Partial Differential Equations

In this paper, we develop a new Best Uniform Rational Approximation-Semi-Discrete (BURA-SD) method taking into account the singularities of the solution of fractional diffusion problems in polygonal domains. The complementary capabilities of the exponential convergence rate of BURA-SD and the $hp$ FEM are explored with the aim of maximizing the overall performance. A challenge here is the emerging singularly perturbed diffusion–reaction equations. The main contributions of this paper include asymptotically accurate error estimates, ending with sufficient conditions to balance errors of different origins, thereby guaranteeing the high computational efficiency of the method. Full article

Simulating surface conditions by solving the wave equation of a sucker-rod string is the theoretical basis of a sucker-rod pumping system. To overcome the shortcomings of the conventional finite difference method and analytical solution, this work describes a novel hybrid method that combines the analytical solution with the finite difference method. In this method, an analytical solution of the tapered rod wave equation with a recursive matrix form based on the Fourier series is proposed, a unified pumping condition model is established, a modified finite difference method is given, a hybrid strategy is established, and a convergence calculation method is proposed. Based on two different types of oil wells, the analytical solutions are verified by comparing different methods. The hybrid method is verified by using the finite difference method simulated data and measured oil data. The pumping speed sensitivity and convergence of the hybrid method are studied. The results show that the proposed analytical solution has high accuracy, with a maximum relative error relative to that of the classical finite difference method of 0.062%. The proposed hybrid method has a high simulation accuracy, with a maximum relative area error relative to that of the finite difference method of 0.09% and a maximum relative area error relative to measured data of 1.89%. Even at higher pumping speeds, the hybrid method still has accuracy. The hybrid method in this paper is convergent. The introduction of the finite difference method allows the hybrid method to more easily converge. The novelty of this work is that it combines the advantages of the finite difference method and the analytical solution, and it provides a convergence calculation method to provide guidance for its application. The hybrid method presented in this paper provides an alternative scheme for predicting the behavior of sucker-rod pumping systems and a new approach for solving wave equations with complex boundary conditions. Full article

This article introduces a finite element method based on the C-Bézier basis function for second-order elliptic equations. The trial function of the finite element method is set up using a combination of C-Bézier tensor product bases. One advantage of the C-Bézier basis is that it has a free shape parameter, which makes geometric modeling more convenience and flexible. The performance of the C-Bézier basis is searched for by studying three test examples. The numerical results demonstrate that this method is able to provide more accurate numerical approximations than the classical Lagrange basis. Full article

In this work, an entropy-stable and well-balanced numerical scheme for a one-dimensional blood flow model is presented. Such a scheme was obtained from an explicit entropy-conservative flux along with a second-order discretisation of the source term by using centred finite differences. We prove that the scheme is entropy-stable and preserves steady-state solutions. In addition, some numerical examples are included to test the performance of the proposed scheme. Full article

We present a modified characteristic finite element method that exhibits second-order spatial accuracy for solving convection–reaction–diffusion equations on surfaces. The temporal direction adopted the backward-Euler method, while the spatial direction employed the surface finite element method. In contrast to regular domains, it is observed that the point in the characteristic direction traverses the surface only once within a brief time. Thus, good approximation of the solution in the characteristic direction holds significant importance for the numerical scheme. In this regard, Taylor expansion is employed to reconstruct the solution beyond the surface in the characteristic direction. The stability of our scheme is then proved. A comparison is carried out with an existing characteristic finite element method based on face mesh. Numerical examples are provided to validate the effectiveness of our proposed method. Full article

Partial differential equations (PDEs) usually apply for modeling complex physical phenomena in the real world, and the corresponding solution is the key to interpreting these problems. Generally, traditional solving methods suffer from inefficiency and time consumption. At the same time, the current rise in machine learning algorithms, represented by the Deep Operator Network (DeepONet), could compensate for these shortcomings and effectively predict the solutions of PDEs by learning the operators from the data. The current deep learning-based methods focus on solving one-dimensional PDEs, but the research on higher-dimensional problems is still in development. Therefore, this paper proposes an efficient scheme to predict the solution of two-dimensional PDEs with improved DeepONet. In order to construct the data needed for training, the functions are sampled from a classical function space and produce the corresponding two-dimensional data. The difference method is used to obtain the numerical solutions of the PDEs and form a point-value data set. For training the network, the matrix representing two-dimensional functions is processed to form vectors and adapt the DeepONet model perfectly. In addition, we theoretically prove that the discrete point division of the data ensures that the model loss is guaranteed to be in a small range. This method is verified for predicting the two-dimensional Poisson equation and heat conduction equation solutions through experiments. Compared with other methods, the proposed scheme is simple and effective. Full article

This paper calculates numerical solutions of an extended three-coupled Korteweg–de Vries system by the q-homotopy analysis transformation method (q-HATM), which is a hybrid of the Laplace transform and the q-homotopy analysis method. Multiple investigations inspecting planetary oceans, optical cables, and cosmic plasma have employed the KdV model, significantly contributing to its development. The uniqueness, convergence, and maximum absolute truncation error of this algorithm are demonstrated. A numerical simulation has been performed to validate the accuracy and validity of the proposed approach. With high accuracy and few algorithmic processes, this algorithm supplies a series solution in the form of a recursive relation. Full article

In recent years, Physics-Informed Neural Networks (PINNs) have drawn great interest among researchers as a tool to solve computational physics problems. Unlike conventional neural networks, which are black-box models that “blindly” establish a correlation between input and output variables using a large quantity of labeled data, PINNs directly embed physical laws (primarily partial differential equations) within the loss function of neural networks. By minimizing the loss function, this approach allows the output variables to automatically satisfy physical equations without the need for labeled data. The Navier–Stokes equation is one of the most classic governing equations in thermal fluid engineering. This study constructs a PINN to solve the Navier–Stokes equations for a 2D incompressible laminar flow problem. Flows passing around a 2D circular particle are chosen as the benchmark case, and an elliptical particle is also examined to enrich the research. The velocity and pressure fields are predicted by the PINNs, and the results are compared with those derived from Computational Fluid Dynamics (CFD). Additionally, the particle drag force coefficient is calculated to quantify the discrepancy in the results of the PINNs as compared to CFD outcomes. The drag coefficient maintained an error within 10% across all test scenarios. Full article

This article is concerned with the convergence properties of the Strang splitting method for the Degasperis-Procesi equation, which models shallow water dynamics. The challenges of analyzing splitting methods for this equation lie in the fact that the involved suboperators are both nonlinear. In this paper, instead of building the second order convergence in $L^2$ for the proposed method directly, we first show that the Strang splitting method has first order convergence in $H^2$. In the analysis, the Lie derivative bounds for the local errors are crucial. The obtained first order convergence result provides the $H^2$ boundedness of the approximate solutions, thereby enabling us to subsequently establish the second order convergence in $L^2$ for the Strang splitting method. Full article

In this paper, an explicit boundary-type numerical procedure, including a constraint-type fictitious time integration method (FTIM) and a two-point boundary solution of the Lie-group shooting method (LGSM), is constructed to tackle nonlinear nonhomogeneous backward heat conduction problems (BHCPs). Conventional methods cannot effectively overcome numerical instability to solve inverse problems that lack initial conditions and take a long time to calculate, even using different variable transformations and regularization techniques. Therefore, an explicit-type numerical procedure is developed from the FTIM and the LGSM to avoid numerical instability and numerical iterations. First, a two-point boundary solution of the LGSM is introduced into the numerical algorithm. Then, the maximum and minimum values of the initial guess value can be determined linearly from the boundary conditions at the initial and final times. Finally, an explicit-type boundary-type numerical procedure, including a boundary value solution and constraint-type FTIM, can directly avoid the numerical iterative problems of BHCPs. Several nonlinear examples are tested. Based on the numerical results shown, this boundary-type numerical procedure using a two-point solution can directly obtain an approximated solution and can achieve stable convergence to boundary conditions, even if numerical iterations occur. Furthermore, the numerical efficiency and accuracy are better than in the previous literature, even with an increased computational time span without the regularization technique. Full article

A novel dimension splitting method is proposed for the efficient numerical simulation of a biochemotaxis model, which is a coupled system of chemotaxis–fluid equations and incompressible Navier–Stokes equations. A second-order pressure correction method is employed to decouple the velocity and pressure for the Navier–Stokes equations. Then, the alternating direction implicit scheme is used to solve the velocity equation, and the operator with dimension splitting effect is used instead of the traditional elliptic operator to solve the pressure equation. For the chemotactic equation, the operator splitting method and extrapolation technique are used to solve oxygen and cell density to achieve second-order time accuracy. The proposed dimension splitting method splits the two-dimensional problem into a one-dimensional problem by splitting the spatial derivative, which reduces the computation and storage costs. Finally, through interesting experiments, we show the evolution of the cell plume shape during the descent process. The effect of changing specific parameters on the velocity and plume shape during the descent process is also studied. Full article

Physics-informed neural networks (PINNs) have been widely used to solve partial differential equations in recent years. But studies have shown that there is a gradient pathology in PINNs. That is, there is an imbalance gradient problem in each regularization term during back-propagation, which makes it difficult for neural network models to accurately approximate partial differential equations. Based on the depth-weighted residual neural network and neural attention mechanism, we propose a new mixed-weighted residual block in which the weighted coefficients are chosen autonomously by the optimization algorithm, and one of the transformer networks is replaced by a skip connection. Finally, we test our algorithms with some partial differential equations, such as the non-homogeneous Klein–Gordon equation, the (1+1) advection–diffusion equation, and the Helmholtz equation. Experimental results show that the proposed algorithm significantly improves the numerical accuracy. Full article

In this paper, we study the stability of the stochastic parabolic differential equation with dependent coefficients. We consider the stability of an abstract Cauchy problem for the solution of certain stochastic parabolic differential equations in a Hilbert space. For the solution of the initial-boundary value problems (IBVPs), we obtain the stability estimates for stochastic parabolic equations with dependent coefficients in specific applications. Full article

This paper proposes a higher-order blended compact difference (BCD) scheme on nonuniform grids for solving the three-dimensional (3D) convection–diffusion equation with variable coefficients. The BCD scheme has fifth- to sixth-order accuracy and considers the first and second derivatives of the unknown function as unknowns as well. Unlike other schemes that require grid transformation, the BCD scheme does not require any grid transformation and is simple and flexible in grid subdivisions. Concurrently, the corresponding high-order boundary schemes of the first and second derivatives have also been constructed. We tested the BCD scheme on three problems that involve convection-dominated and boundary-layer features. The numerical results show that the BCD scheme has good adaptability and high resolution on nonuniform grids. It outperforms the BCD scheme on uniform grids and the high-order compact scheme on nonuniform grids in the literature in terms of accuracy and resolution. Full article

In this paper, a thermomechanical problem involving a viscoelastic Timoshenko beam is analyzed from a numerical point of view. The so-called thermodiffusion effects are also included in the model. The problem is written as a linear system composed of two second-order-in-time partial differential equations for the transverse displacement and the rotational movement, and two first-order-in-time partial differential equations for the temperature and the chemical potential. The corresponding variational formulation leads to a coupled system of first-order linear variational equations written in terms of the transverse velocity, the rotation speed, the temperature and the chemical potential. The existence and uniqueness of solutions, as well as the energy decay property, are stated. Then, we focus on the numerical approximation of this weak problem by using the implicit Euler scheme to discretize the time derivatives and the classical finite element method to approximate the spatial variable. A discrete stability property and some a priori error estimates are shown, from which we can conclude the linear convergence of the approximations under suitable additional regularity conditions. Finally, some numerical simulations are performed to demonstrate the accuracy of the scheme, the behavior of the discrete energy decay and the dependence of the solution with respect to some parameters. Full article

Reflected partial differential equations (PDEs) have important applications in financial mathematics, stochastic control, physics, and engineering. This paper aims to present a numerical method for solving high-dimensional reflected PDEs. In fact, overcoming the “dimensional curse” and approximating the reflection term are challenges. Some numerical algorithms based on neural networks developed recently fail in solving high-dimensional reflected PDEs. To solve these problems, firstly, the reflected PDEs are transformed into reflected backward stochastic differential equations (BSDEs) using the reflected Feyman–Kac formula. Secondly, the reflection term of the reflected BSDEs is approximated using the penalization method. Next, the BSDEs are discretized using a strategy that combines Euler and Crank–Nicolson schemes. Finally, a deep neural network model is employed to simulate the solution of the BSDEs. The effectiveness of the proposed method is tested by two numerical experiments, and the model shows high stability and accuracy in solving reflected PDEs of up to 100 dimensions. Full article

Heat transfer and fluid dynamics modeling in porous media is a widely explored topic in physics and applied mathematics, and it involves advanced numerical methods to address its non-linear nature. One interesting application has been the urban-heat-island (UHI) numerical simulation. The UHI is a negative consequence of the increasing urbanization in cities, which is defined as the presence of warm temperatures inside the urban canopy in contrast to the colder surroundings. Furthermore, an interesting phenomena occurs within a UHI context when the city transitions from a heat island to a cold island, matching the increases and decreases of solar radiation over the span of a day, as well as the decrease in the UHI intensity as a result of wind action. The numerical study in this paper had, as its main goal, to reproduce this phenomenon. Therefore, the key elements proposed in this work were the following: A 2D horizontal urban–rural domain that had a variable porosity with a Gaussian distribution centered in the city center. A non-stationary Darcy–Forchheimer–Brinkman model to simulate the flow in porous media, combined with an air–soil heat transport model linked by a balancing equation for the surface energy that includes the evapotranspiration of plants. In regards to the numerical resolution of the model, a classical numerical methodology based on the finite elements of Lagrange P₁ type combined with explicit and implicit time-marching schemes have been effective for high-quality numerical simulations. Several numerical tests were performed on a domain inspired by the metropolitan region of Guadalajara (Mexico), in which not only the temperature inversion was reproduced but also the simulation of the UHI transition by strong wind gusts. Full article

The current research study is focusing on the investigation of the physical effects of thermal radiation on heat and mass transfer of a nanofluid located around a sphere. The configuration is investigated by solving the partial differential equations governing the phenomenon. By using suitable non-dimensional variables, the governing set of partial differential equations is transformed into a dimensionless form. For numerical simulation, the attained set of dimensionless partial differential equations is discretized by using the finite difference method. The effects of the governing parameters, such as the Brownian motion parameter, the thermophoresis parameter, the radiation parameter, the Prandtl number, and the Schmidt number on the velocity field, temperature distribution, and mass concentration, are presented graphically. Moreover, the impacts of these physical parameters on the skin friction coefficient, the Nusselt number, and the Sherwood number are displayed in the form of tables. Numerical outcomes reflect that the effects of the radiation parameter, thermophoresis parameter, and the Brownian motion parameter intensify the profiles of velocity, temperature, and concentration at different circumferential positions on the sphere. Full article

In this paper, we propose a finite volume element method of primal-dual type to solve the ill-posed elliptic problem, that is, the elliptic problem with lacking or overlapping boundary value condition. We first establish the primal-dual finite volume element scheme by introducing the Lagrange multiplier $\lambda$ and prove the well-posedness of the discrete scheme. Then, the error estimations of the finite volume solution are derived under some proper norms including the $H^1$-norm. Numerical experiments are provided to verify the effectiveness of the proposed finite volume element method at last. Full article

We herein mainly employ a proper orthogonal decomposition (POD) to study the reduced dimension of unknown solution coefficient vectors in the Crank–Nicolson finite element (FE) (CNFE) method for the symmetric tempered fractional diffusion equation so that we can build the reduced-dimension recursive CNFE (RDRCNFE) method. In this case, the RDRCNFE method keeps the same basic functions and accuracy as the CNFE method. Especially, we adopt the matrix analysis to discuss the stability and convergence of RDRCNFE solutions, resulting in the very laconic theoretical analysis. We also use some numerical simulations to confirm the correctness of theoretical results. Full article

In this paper, we consider a fully discrete modular grad-div stabilization algorithm for time-dependent thermally coupled magnetohydrodynamic (MHD) equations. The main idea of the proposed algorithm is to add an extra minimally intrusive module to penalize the divergence errors of velocity and improve the computational efficiency for increasing values of the Reynolds number and grad-div stabilization parameters. In addition, we provide the unconditional stability and optimal convergence analysis of this algorithm. Finally, several numerical experiments are performed and further indicated these advantages over the algorithm without grad-div stabilization. Full article