Nonsmooth Solution Research Articles

ROBUST IMPLICIT DIFFERENCE APPROACH FOR THE TIME-FRACTIONAL KURAMOTO–SIVASHINSKY EQUATION WITH THE NON-SMOOTH SOLUTION

This paper formulates the L1 implicit difference scheme (L1IDS) for the time-fractional Kuramoto–Sivashinsky equation (TFKSE) with non-smooth solution. The TFKSE is one of useful descriptions for modeling flame-propagation, viscous flow problems, and reaction–diffusion systems. The proposed method approximates the unknown solution by using two main stages. At the first stage, the L1 method with nonuniform meshes and the general centered difference method is adopted to discretize the Caputo fractional derivative and the spatial derivative, respectively. In the second stage, the fully-discrete L1IDS is established with the help of the Galerkin scheme based on piecewise linear test functions. Meanwhile, an iterative algorithm is adopted to solve the nonlinear systems. Furthermore, the convergence and stability of the proposed method are both demonstrated and confirmed numerically. Finally, three numerical examples highlight the accuracy and efficiency of the proposed strategy.

A new sixth-order immersed interface method for solving Poisson equations with straight interfaces

This paper introduces a sixth-order Immersed Interface Method (IIM) for addressing 2D Poisson problems characterized by a discontinuous forcing function with straight interfaces. In the presence of this discontinuity, the problem exhibits a non-smooth solution at the interface that divides the domain into two regions. Here, the IIM is employed to compute the solution on a fixed Cartesian grid. This method integrates necessary jump conditions resulting from the interface into the numerical schemes. In order to achieve a sixth-order method, the proposed approach combines implicit finite differences with the IIM. The proposed scheme is efficient because the matrix arising from discretization remains the same as in the smooth problem, and changes are made to the resulting linear system by introducing new terms on the right side. These supplementary terms account for the discontinuities in the solution and its derivatives, with calculations restricted near the interface. The paper demonstrates the accuracy of the proposed method through various numerical examples.

Neural Eikonal solver: Improving accuracy of physics-informed neural networks for solving eikonal equation in case of caustics

The concept of physics-informed neural networks has become a useful tool for solving differential equations due to its flexibility. A few approaches use this concept to solve the eikonal equation that describes the first-arrival traveltimes of waves propagating in smooth heterogeneous velocity models. However, the challenge of the eikonal is exacerbated by the velocity models producing caustics, resulting in instabilities and deterioration of accuracy due to the non-smooth solution behavior. In this paper, we revisit the problem of solving the eikonal equation using neural networks to tackle caustic pathologies. We introduce the novel Neural Eikonal Solver (NES) for solving the isotropic eikonal equation in two formulations: the one-point problem is for a fixed source location; the two-point problem is for an arbitrary source-receiver pair. We present several techniques which provide stability in the case of caustics: improved factorization; non-symmetric loss function based on Hamiltonian; gaussian activation; symmetrization. In our tests, NES showed the relative mean-absolute error of 0.2-0.4% from the second-order factored Fast Marching Method with a similar inference time, and outperformed existing neural-network solvers giving 10-60 times lower errors and 2-30 times faster training. The one-point NES provides the most accurate solution, while the two-point NES gives slightly lower accuracy but an extremely compact representation with all spatial derivatives. It can be useful in many seismic problems: massive computations of traveltimes for millions of source-receiver pairs in Kirchhoff migration; modeling of ray amplitudes using spatial derivatives; traveltime tomography; earthquake localization; ray multipathing analysis.

Superconvergence analysis of anisotropic finite element method for the time fractional substantial diffusion equation with smooth and nonsmooth solutions

In this work, anisotropic bilinear finite element and second‐order temporal approximation are adopted to establish a fully discrete scheme for the fractional substantial diffusion equation with variable coefficient. We prove the proposed discrete scheme is unconditionally stable in the senses of ‐norm and ‐norm. By introducing a new projection, the optimal convergence error in ‐norm and the superclose property in ‐norm are obtained under the condition of , where is the exact solution of the problem. Through interpolation post processing technique, we arrive at the global superconvergence of the interpolation. Furthermore, an improved algorithm is proposed to solve the problem with nonsmooth solution. Finally, numerical tests are given to illustrate the validity and efficiency of our theoretical analysis.

A high-order numerical scheme based on graded mesh and its analysis for the two-dimensional time-fractional convection-diffusion equation

This work proposes a numerical method for solving the two-dimensional time-fractional convection-diffusion (TFCD) equation. The time-fractional derivative is considered in the sense of Caputo. The considered problem with Caputo temporal fractional derivative generally shows a weak singularity at the initial time. In order to handle the weak singularity at t = 0 , the time-fractional derivative is discretized on a non-uniform graded mesh. A high-order compact operator is employed for approximating the derivatives in space directions. The resulting system of equations is then solved using the Alternating Direction Implicit (ADI) approach. We prove the stability and convergence of this scheme. Numerical experiment is performed to demonstrate the applicability and accuracy of the method. The considered numerical example subjected to nonsmooth solution confirms that the method has an order min ⁡ ( 2 − α , r α , 2 α + 1 ) in the temporal direction, where α ∈ ( 0 , 1 ) is the order of the fractional derivative and r is a parameter used in construction of the graded mesh and has a rate of convergence of order four in spatial direction. The proposed method on graded mesh has an advantage in terms of numerical accuracy over the method on the uniform mesh. The CPU time (in seconds) for the present method is also provided.

A space-time spectral approximation for solving nonlinear variable-order fractional convection-diffusion equations with nonsmooth solutions

One of the problems in the numerical analysis of solutions is the nonlinear variable-order fractional convection-diffusion equations for nonsmooth solutions. We offer a numerical technique based on the shifted Legendre Gauss-Lobatto collocation and the shifted Chebyshev Gauss-Radau collocation to solve the problem. The technique with shifted Legendre Gauss-Lobatto and shifted Chebyshev Gauss-Radau nodes is applied to diminish nonlinear variable-order fractional convection-diffusion equations to an easily-solvable system of algebraic equations. Besides, we give numerical test examples to show that the approach can preserve the nonsmooth solution of the underlying problems.

High-Order Multivariate Spectral Algorithms for High-Dimensional Nonlinear Weakly Singular Integral Equations with Delay

One of the open problems in the numerical analysis of solutions to high-dimensional nonlinear integral equations with memory kernel and proportional delay is how to preserve the high-order accuracy for nonsmooth solutions. It is well-known that the solutions to these equations display a typical weak singularity at the initial time, which causes challenges in developing high-order and efficient numerical algorithms. The key idea of the proposed approach is to adopt a smoothing transformation for the multivariate spectral collocation method to circumvent the curse of singularity at the beginning of time. Therefore, the singularity of the approximate solution can be tailored to that of the exact one, resulting in high-order spectral collocation algorithms. Moreover, we provide a framework for studying the rate of convergence of the proposed algorithm. Finally, we give a numerical test example to show that the approach can preserve the nonsmooth solution to the underlying problems.

Automatic balancing parameter selection for Tikhonov-TV regularization

This paper considers large-scale linear ill-posed inverse problems whose solutions can be represented as sums of smooth and piecewise constant components. To solve such problems we consider regularizers consisting of two terms that must be balanced. Namely, a Tikhonov term guarantees the smoothness of the smooth solution component, while a total-variation (TV) regularizer promotes blockiness of the non-smooth solution component. A scalar parameter allows to balance between these two terms and, hence, to appropriately separate and regularize the smooth and non-smooth components of the solution. This paper proposes an efficient algorithm to solve this regularization problem by the alternating direction method of multipliers (ADMM). Furthermore, a novel algorithm for automatic choice of the balancing parameter is introduced, using robust statistics. The proposed approach is supported by some theoretical analysis, and numerical experiments concerned with different inverse problems are presented to validate the choice of the balancing parameter.

Fast high-order compact difference scheme for the nonlinear distributed-order fractional Sobolev model appearing in porous media

In this article, we present an efficient numerical algorithm, which combines the fourth-order compact difference scheme (CDS) in space with the fast time two-mesh (TT-M) FBN-θ method, to solve the nonlinear distributed-order fractional Sobolev model appearing in porous media. We also construct the corrected CDS by adding the starting part to deal with the problem with nonsmooth solution and recovery the convergence rate. We derive optimal convergence results and prove the stability of the presented numerical algorithm. Finally, we implement numerical experiments by taking several examples with smooth and nonsmooth solutions to verify the correctness of the theoretical results, the effectiveness of the corrected algorithm and the computational efficiency of the fast algorithm.

Analytical and numerical investigation on the tempered time-fractional operator with application to the Bloch equation and the two-layered problem

In the continuous-time random walk model, the time-fractional operator usually expresses an infinite waiting time probability density. Different from that usual setting, this work considers the tempered time-fractional operator, which reflects a finite waiting time probability density. Firstly, we analyse the solution of a tempered benchmark problem, which shows a weak singularity near the initial time. The L1 scheme on graded mesh and the weighted shifted Grünwald–Letnikov formula with correction terms are adapted to deal with the non-smooth solution, in which we compare these two methods systematically in terms of the convergence and consumed CPU time. Furthermore, a fast calculation for the time tempered Caputo fractional derivative is developed based on a sum-of-exponentials approximation, which significantly reduces the running time. Moreover, the tempered operator is applied to the Bloch equation in nuclear magnetic resonance and a two-layered problem with composite material exhibiting distinct memory effects, for which both the analytical (or semi-analytical) and numerical solutions are derived using transform techniques and finite difference methods. Data-fitting results verify that the tempered time-fractional model is much effective to describe the MRI data. An important finding is that, compared with the fractional index, the tempered operator parameter could further accelerate the diffusion. The tempered model with two parameters alpha and rho are more flexible, which can avoid choosing a too small fractional index leading to low regularity and strong heterogeneity.

An alternating direction implicit legendre spectral method for simulating a 2D multi-term time-fractional Oldroyd-B fluid type diffusion equation

In this paper, an alternating direction implicit legendre spectral (ADILS) method is developed for the two-dimensional (2D) multi-term time fractional Oldroyd-B fluid type diffusion equation. This method is provided based on legendre spectral approximation in space and a new difference discretization in time, and it can convert 2D computation to two 1D computations. Meanwhile, a very important lemma is proposed to prove the unconditional stability and convergence of this ADILS scheme. Especially, the corrected scheme is utilized to increase the convergence accuracy for the non-smooth solution. With three numerical experiments, the validity of the method is confirmed. The present method also can be extended to solve the generalized non-Newtonian fluid type equations, multi-term time fractional mixed diffusion and diffusion-wave equations.

Pointwise error estimate and stability analysis of fourth-order compact difference scheme for time-fractional Burgers’ equation

A novel implicit high-order compact difference scheme is established for the time-fractional Burgers’ equation based on a newly developed nonlinear compact operator and the reduction order technique under the uniform mesh and graded mesh. Uniqueness, boundedness, convergence in the pointwise sense and stability in L2-norm of the discrete solutions are proved by the Browder fixed point theorem and energy argument. Numerical examples with the smooth solution and nonsmooth solution are provided to confirm numerical theoretical results and demonstrate the efficiency of the high-order compact difference scheme.

Stable numerical schemes for time-fractional diffusion equation with generalized memory kernel

The paper aims to develop the stable numerical schemes for generalized time-fractional diffusion equations (GTFDEs) with smooth and non-smooth solutions on the non-uniform grid. In time, the generalized Caputo derivative is discretized by a difference scheme of order (2−α) on a non-uniform grid where 0<α<1. Choosing the non-uniform meshes in the case of the smooth and non-smooth solution is also essential, so we graded the mesh in both cases separately. Stability and convergence for smooth as well as non-smooth solutions are obtained in L2-norm and L∞-norm respectively. Several numerical results are presented to show how the grading of meshes is essential. Also, numerical results validate the efficiency and effectiveness of proposed schemes and show how a non-uniform grid produces better results.

Traveling nonsmooth solution and conserved quantities of long nonlinear internal waves

Traveling nonsmooth solution and conserved quantities of long nonlinear internal waves

Highly accurate solutions for space–time fractional Schrödinger equations with non-smooth continuous solution using the hybrid clique functions

Highly accurate solutions for space–time fractional Schrödinger equations with non-smooth continuous solution using the hybrid clique functions

An extrapolated finite difference method for two-dimensional fractional boundary value problems with non-smooth solution

In this paper, the well-known shifted Grünwald–Letnikov formula is revisited to solve the two-dimensional fractional boundary value problems (FBVPs) with non-smooth solution. Both stability analysis and error estimates of the scheme are carried out in the maximum norm. An improved algorithm is presented based on the extrapolation technique to obtain high-order accuracy for the problems with low regularity. Various numerical experiments are given to support the theoretical finding and verify the effectiveness of the improved algorithm. It is illustrated that, by using the proposed algorithm, both accuracy and convergence rate can be significantly improved for solving the two-dimensional FBVPs as well as fractional diffusion equations with non-smooth solution. Especially, the convergence rate of corrected numerical solutions even can reach second-order in the case of solving problems involved one-sided fractional derivatives.

Numerical solutions for fractional initial value problems of distributed-order

This paper addresses spectral collocation techniques to treat with the fractional initial value problem of distributed-order. We introduce three algorithms based on shifted fractional order Jacobi orthogonal functions outputted by Jacobi polynomials. The shifted fractional order Jacobi–Gauss–Radau collocation method is developed for approximating the fractional initial value problem of distributed-order. The principal target in our techniques is to transform the fractional initial value problem of distributed-order to a system of algebraic equations. Some numerical examples are given to test the accuracy and applicability of our technique. It is known that the accuracy of numerical approaches for nonsmooth solution is deteriorated. Employing fractional order Jacobi functions instead of the classical Jacobi stopped this deterioration.

Memetic algorithm with non-smooth penalty for capacitated arc routing problem

The capacitated arc routing problem (CARP) has attracted much attention during last decades due to its wide applications. However, the existing research methods still have a little potential to make full use of the characteristics of CARP. This paper aims to mine the essential characteristics of arc routing problem that node routing problem does not have. Based on the observation on characteristics of arc routing instances, smooth condition is proposed and constructed as a rule to divide the link between two tasks into smooth link and non-smooth link. Then smooth degree is defined to measure the influence of non-smooth links on solution and a small smooth degree means the better quality for a solution. The effect of smooth degree is verified through simulation comparison on several instance sets, which indicates that there is a positive correlation between smooth degree and the total cost of a solution. Non-smooth penalty is used to drive the non-smooth solution to smooth and to improve its total cost. Then non-smooth penalty is inserted into path-scanning variants and new construction algorithms are obtained. A partial reconstruction method (PRM) is designed using these construction algorithms as an alternative kernel method. In order to further reduce the routes number, a reinsert method (RiM) is proposed. Combining these two methods with traditional local search algorithms, a memetic algorithm with non-smooth penalty (MANSP) is proposed which is originated from the initial observation on the essential characteristics of arc routing problem. Extensive experiments on smooth degree, penalty factor, and comparison with state-of-the-art algorithms show that the proposed strategies are effective and the proposed algorithm MANSP performs very competitive.

Reliable Residual-Based Error Estimation for the Finite Cell Method

In this work, the reliability of a residual-based error estimator for the Finite Cell method is established. The error estimator is suitable for the application of hp-adaptive finite elements and allows for Neumann boundary conditions on curved boundaries. The reliability proof of the error estimator relies on standard arguments of residual-based a posteriori error control, but includes several modifications with respect to the error contributions associated with the volume residuals as well as the jumps across inner edges and Neumann boundary parts. Important ingredients of the proof are Stein’s extension theorem and a modified trace theorem which estimates the norm of the trace on (curved) boundary parts in terms of the local mesh size and polynomial degree. The efficiency of the error estimator is also considered by discussing an artificial example which yields an efficiency index depending on the mesh-family parameter h. Numerical experiments on more realistic domains, however, suggest global efficiency with the occurrence of a large overestimation on only few cut elements. In the experiments the reliability of the error estimator is demonstrated for h- and p-uniform as well as for hp-geometric and h-adaptive refinements driven by the error estimator. The practical applicability of the error estimator is also studied for a 3D problem with a non-smooth solution.

A fast finite difference/RBF meshless approach for time fractional convection-diffusion equation with non-smooth solution

In this paper, we propose a fast finite difference/RBF meshless method for time fractional convection-diffusion equation with singularity at the initial moment. Firstly, the time fractional derivative term is discretized by classical L1 format on graded meshes, and the spatial derivative term is approximated by meshless method. The meshless method does not need to construct meshes, which is more conducive for dealing with complex regions or complex boundary conditions. In addition, we adopt proper orthogonal decomposition (POD) technology to derive reduced-order RBF meshless extrapolation algorithm, which effectively reduces the amount of calculation and improves the calculation efficiency. Finally, the accuracy and efficiency of the method are verified by numerical examples.