Non-smooth Solutions Research Articles

A note on exponential Rosenbrock–Euler method for the finite element discretization of a semilinear parabolic partial differential equation

In this paper, we consider the numerical approximation of a general second order semilinear parabolic partial differential equation. Equations of this type arise in many contexts, such as transport in porous media. Using finite element method for space discretization and the exponential Rosenbrock–Euler method for time discretization, we provide a convergence proof in space and time under only the standard Lipschitz condition of the nonlinear part, for both smooth and nonsmooth initial solutions. This is in contrast to restrictive assumptions made in the literature, where the authors have considered only approximation in time so far in their convergence proofs. The main result reveals how the convergence orders in both space and time depend heavily on the regularity of the initial data. In particular, the method achieves optimal convergence order Oh2+Δt2 when the initial data belongs to the domain of the linear operator. Numerical simulations to sustain our theoretical result are provided.

Finite Difference Method for Two-Dimensional Nonlinear Time-Fractional Subdiffusion Equation

Abstract In this article, we propose an implicit-explicit scheme combining with the fast solver in space to solve two-dimensional nonlinear time-fractional subdiffusion equation. The applications of implicit-explicit scheme and fast solver will smartly enhance the computational efficiency. Due to the non-smoothness (or low regularities) of solutions to fractional differential equations, correction terms are introduced in the proposed scheme to improve the accuracy of error. The stability and convergence of the present scheme are also investigated. Numerical examples are carried out to demonstrate the efficiency and applicability of the derived scheme for both linear and nonlinear fractional subdiffusion equations with non-smooth solutions.

Galerkin and Collocation Methods for Weakly Singular Fractional Integro-differential Equations

This paper describes the collocation and Galerkin’s approaches for fractional integro-differential equations (FIDEs). We explain the application of Jacobi polynomials to solve the FIDEs which convert the problem into a system of algebraic equations. To approximate the solution of FIDEs by Jacobi polynomials, a suitable variable transformation is applied which assures that the solution of the transformed FIDEs is sufficiently smooth. This results in a rapid convergence of both the methods with Jacobi polynomials even when the solution is not smooth. The error estimate and convergence analysis for presented numerical methods are provided. To perform the numerical simulations, two test examples (linear and nonlinear) are considered with non-smooth solutions, and numerical results are presented. Further, the comparative study of the presented schemes with some existing numerical schemes is provided.

A new reproducing kernel collocation method for nonlocal fractional boundary value problems with non-smooth solutions

In this letter, a new reproducing kernel collocation method is presented for solving nonlocal fractional boundary value problems with non-smooth solutions. The method is based on the reproducing kernel basis function and the polynomial basis function. The present method can avoid finding a reproducing kernel satisfying nonlocal boundary conditions. The effectiveness of the numerical scheme is illustrated through two numerical examples.

Wave propagation in elastic trusses: An approach via retarded potentials

We propose a space–time boundary element method for the dynamic simulation of elastic truss systems. The considered truss systems consist of several members, where in each elastic rod the dynamic behaviour is governed by the 1D wave equation. The time domain fundamental solution and boundary integral equations are used to establish the dynamic Dirichlet-to-Neumann map for a single rod. Thus, we are able to reduce the problem to the nodes of the truss system and therefore only a temporal discretization at the truss nodes is necessary. We introduce a stepwise solution strategy with local step size which ensures stability. Furthermore, the discretization within each of these time steps can be refined adaptively to reduce the approximation error efficiently. The optimal convergence of the method is demonstrated in numerical examples. Due to adaptive refinement, this optimal convergence rate is retained even for non-smooth solutions. Finally, the method is applied to study typical truss systems.

A projection‐based algorithm for ℓ2‐ℓp Tikhonov regularization

This paper deals with the minimization problem associated with Tikhonov regularization where the penalty term is a ℓp‐norm with p ∈ (1,2). An algorithm based on fixed‐point iterations is proposed, and theoretical convergence results are given. Further, combining projection over the Krylov subspace generated by the Golub‐Kahan bidiagonalization procedure and the proposed iterative algorithm, we derive an iterative procedure that is well suited for large‐scale ℓ2‐ℓp Tikhonov regularization. The performance and efficiency of the proposed algorithms on image restoration problems like 2D tomography and super resolution, as well as on classical discrete ill‐posed problems having solutions with flat and smooth parts or nonsmooth solutions are also illustrated.

A Finite-Volume Approach to 1D Nonlinear Elastic Waves: Application to Slow Dynamics

A numerical method for longitudinal wave propagation in nonlinear elastic solids is presented. Here, we consider polynomial stress-strain relationships, which are widely used in nondestructive evaluation. The large-strain and infinitesimal-strain constitutive laws deduced from Murnaghan's law are detailed , and polynomial expressions are obtained. The Lagrangian equations of motion yield a hyperbolic system of conservation laws. The latter is solved numerically using a finite-volume method with flux limiters based on Roe linearization. The method is tested on the Riemann problem, which yields nonsmooth solutions. The method is then applied to a continuum model with one scalar internal variable, accounting for the softening of the material (slow dynamics).

On Jensen-type Inequalities for Nonsmooth Radial Scattering Solutions of a Loglog Energy-Supercritical Schrödinger Equation

Abstract We prove scattering of solutions of the loglog energy-supercritical Schrödinger equation $i \partial _{t} u + \triangle u = |u|^{\frac{4}{n-2}} u g(|u|)$ with $g(|u|) := \log ^{\gamma } {( \log{(10+|u|^{2})} )}$, $0 < \gamma < \gamma _{n}$, n ∈ {3, 4, 5}, and with radial data $u(0) := u_{0} \in \tilde{H}^{k}:= \dot{H}^{k} (\mathbb{R}^{n})\,\cap\,\dot{H}^{1} (\mathbb{R}^{n})$, where $\frac{n}{2} \geq k> 1 \left(\text{resp.}\,\frac{4}{3}> k > 1\right)$ if n ∈ {3, 4} (resp. n = 5). The proof uses concentration techniques (see e.g., [ 2, 12]) to prove a long-time Strichartz-type estimate on an arbitrarily long time interval J depending on an a priori bound of some norms of the solution, combined with an induction on time of the Strichartz estimates in order to bound these norms a posteriori (see e.g., [ 8, 10]). We also revisit the scattering theory of solutions with radial data in $\tilde{H}^{k}$, $k> \frac{n}{2}$, and n ∈ {3, 4}; more precisely, we prove scattering for a larger range of $\gamma$ s than in [ 10]. In order to control the barely supercritical nonlinearity for nonsmooth solutions, that is, solutions with data in $\tilde{H}^{k}$, $k \leq \frac{n}{2}$, we prove some Jensen-type inequalities.

Fully discrete spectral methods for solving time fractional nonlinear Sine–Gordon equation with smooth and non-smooth solutions

We consider the initial boundary value problem of the time fractional nonlinear Sine–Gordon equation and the fractional derivative is described in Caputo sense with the order α(1 < α < 2). Two fully discrete schemes are developed based on Legendre spectral approximation in space and finite difference discretization in time for smooth solutions and non-smooth solutions, respectively. Numerical stability and convergence are analysed. Numerical experiments for both the fully discrete schemes are presented to confirm our theoretical analysis.

A sparse-grid isogeometric solver

Isogeometric Analysis (IGA) typically adopts tensor-product splines and NURBS as a basis for the approximation of the solution of PDEs. In this work, we investigate to which extent IGA solvers can benefit from the so-called sparse-grids construction in its combination technique form, which was first introduced in the early 90s in the context of the approximation of high-dimensional PDEs.The tests that we report show that, in accordance to the literature, a sparse-grid construction can indeed be useful if the solution of the PDE at hand is sufficiently smooth. Sparse grids can also be useful in the case of non-smooth solutions when some a-priori knowledge on the location of the singularities of the solution can be exploited to devise suitable non-equispaced meshes. Finally, we remark that sparse grids can be seen as a simple way to parallelize pre-existing serial IGA solvers in a straightforward fashion, which can be beneficial in many practical situations.

An improved tau method for the multi-dimensional fractional Rayleigh–Stokes problem for a heated generalized second grade fluid

We develop efficient algorithms based on the Legendre-tau approximation for one- and two-dimensional fractional Rayleigh–Stokes problems for a generalized second-grade fluid. The time fractional derivative is described in the Riemann–Liouville sense. Discussions on the L2-convergence of the proposed method are presented. Numerical results for one- and two-dimensional examples with smooth and nonsmooth solutions are provided to verify the validity of the theoretical analysis, and to illustrate the efficiency of the proposed algorithms.

METHOD FOR NUMERICAL DIFFERENTIATION OF NOISY DATA WITH OUTLIERS

Context. Using of conventional methods of numerical differentiation to the noisy data with outliers leads to significant errors. The objectof this study is the process of numerical differentiation of such data. Objective. The goal of this work is the development of a method of numerical differentiation of the noisy data with outliers to obtain a smooth approximation of the first derivative of original data as well as a smooth approximation of the original data themselves. Method. The proposed method of numerical differentiation is based on solving the problem of minimizing the smoothing functional, which is built on the criteria of a minimum of extent of the solution residual and of an energy constraint of the first derivative of solution. The minimum-extent criterion defines the main part of functional and ensures its stable behavior with respect to the additive noise and outliers. The energy constraint defines the stabilizing part of the functional and provides a smooth solution of the problem. The contribution of these parts is controlled by a regularization parameter. Since the main part of smoothing functional is not convex, then the minimization problem is the non-convex nonlinear programming problem. For the numerical solution of this problem the conjugate gradient method is used. In this method the step size along the descent direction is defined on the set of test steps. These steps minimize the individual components of the main and stabilizing parts of the smoothing functional that allows to move from the one local minimum of the functional to another deeper local minimum. Results. Simulation of the problem of numerical differentiation of noisy data with outliers and processing of the experimental data, which are photoluminescence spectra with narrow line components in their compositions, confirmed the performance of the proposed method. Conclusions. The proposed method can be used for numerical differentiation of noisy data with outliers. It provides a smooth approximation of the first derivative of the original data, as well as a smooth approximation of the original data themselves. This method can be generalized to the case of non-smooth solutions by constructing a stabilizing part of the functional based on the criterion of minimum total variation.

A High Order Conservative Semi-Lagrangian Discontinuous Galerkin Method for Two-Dimensional Transport Simulations

In this paper, we develop a class of high order conservative semi-Lagrangian (SL) discontinuous Galerkin methods for solving multi-dimensional linear transport equations. The methods rely on a characteristic Galerkin weak formulation, leading to $$L^2$$ stable discretizations for linear problems. Unlike many existing SL methods, the high order accuracy and mass conservation of the proposed methods are realized in a non-splitting manner. Thus, the detrimental splitting error, which is known to significantly contaminate long term transport simulations, will be not incurred. One key ingredient in the scheme formulation, borrowed from CSLAM (Lauritzen et al. in J Comput Phys 229(5):1401–1424, 2010), is the use of Green’s theorem which allows us to convert volume integrals into a set of line integrals. The resulting line integrals are much easier to approximate with high order accuracy, hence facilitating the implementation. Another novel ingredient is the construction of quadratic curves in approximating sides of upstream cell, leading to quadratic-curved quadrilateral upstream cells. Formal third order accuracy is obtained by such a construction. The desired positivity-preserving property is further attained by incorporating a high order bound-preserving filter. To assess the performance of the proposed methods, we test and compare the numerical schemes with a variety of configurations for solving several benchmark transport problems with both smooth and nonsmooth solutions. The efficiency and efficacy are numerically verified.

Second-order numerical methods for multi-term fractional differential equations: Smooth and non-smooth solutions

Starting with the asymptotic expansion of the error equation of the shifted Grünwald–Letnikov formula, we derive a new modified weighted shifted Grünwald–Letnikov (WSGL) formula by introducing appropriate correction terms. We then apply one special case of the modified WSGL formula to solve multi-term fractional ordinary and partial differential equations, and we prove the linear stability and second-order convergence for both smooth and non-smooth solutions when the regularity of the solutions is known. We show theoretically and numerically that numerical solutions with good accuracy can be obtained with only a few correction terms. Moreover, the correction terms can be tuned according to the fractional derivative orders without explicitly knowing the analytical solutions. Numerical simulations verify the theoretical results and demonstrate that the new WSGL formula leads to better performance compared to other known numerical approximations with similar resolution.

Veiled Singularities for the Spherically Symmetric Massless Einstein–Vlasov System

This paper continues the investigation of the formation of naked singularities in the collapse of collisionless matter initiated in [RV]. There the existence of certain classes of non-smooth solutions of the Einstein-Vlasov system was proved. Those solutions are self-similar and hence not asymptotically flat. To obtain solutions which are more physically relevant it makes sense to attempt to cut off these solutions in a suitable way so as to make them asymptotically flat. This task, which turns out to be technically challenging, will be carried out in this paper. [RV] A. D. Rendall and J. J. L. Vel\'{a}zquez, A class of dust-like self-similar solutions of the massless Einstein-Vlasov system. Annales Henri Poincare 12, 919-964, (2011).

Classification of traveling wave solutions to the Green–Naghdi model

In this paper, the Green–Naghdi model is investigated by employing the qualitative method. We classify all traveling wave solutions to this model in specified parameter region of the parameter space. Especially, we study the limiting behavior of all smooth and non-smooth periodic solutions as the parameters tend to some special values. Based on the qualitative results, all exact traveling wave solutions as well as their profiles are also given.

Simulating Magnetohydrodynamic Instabilities with Conservative Perturbed MHD Model Using Discontinuous Galerkin Method

AbstractIn magnetically confined plasma research, the understandings of small and large perturbations at equilibrium are both critical for plasma controlling and steady state operation. Numerical simulations using original MHD model can hardly give clear picture for small perturbations, while non-conservative perturbed MHD model may break conservation law, and give unphysical results when perturbations grow large after long-time computation. In this paper, we present a nonlinear conservative perturbed MHD model by splitting primary variables in original MHD equations into equilibrium part and perturbed part, and apply an approach in the framework of discontinuous Galerkin (DG) spatial discretization for numerical solutions. This enables high resolution of very small perturbations, and also gives satisfactory non-smooth solutions for large perturbations, which are both broadly concerned in magnetically confined plasma research. Numerical examples demonstrate satisfactory performance of the proposed model clearly. For small perturbations, the results have higher resolution comparing with the original MHD model; for large perturbations, the non-smooth solutions match well with existing references, confirming reliability of the model for instability investigations in magnetically confined plasma numerical research.

Convergence rates for solving elliptic boundary value problems with singular parameterizations in isogeometric analysis

In this paper, we present convergence rates for solving elliptic boundary value problems with singular parameterizations in isogeometric analysis. First, the approximation errors with the L2(Ω)-norm and the H1(Ω)-seminorm are estimated locally. The impact of singularities is considered in this framework. Second, the convergence rates for solving PDEs with singular parameterizations are discussed. These results are based on a weak solution space that contains all of the weak solutions of elliptic boundary value problems with smooth coefficients. For the smooth weak solutions obtained by isogeometric analysis with singular parameterizations and the finite element method, both are shown to have the optimal convergence rates. For non-smooth weak solutions, the optimal convergence rates are reached by setting proper singularities of a controllable parameterization, even though convergence rates are not optimal by finite element method, and the convergence rates by isogeometric analysis with singular parameterizations are better than the ones by the finite element method.

A hybrid pressure–density-based Mach uniform algorithm for 2D Euler equations on unstructured grids by using multi-moment finite volume method

We propose a novel Mach-uniform numerical model for 2D Euler equations on unstructured grids by using multi-moment finite volume method. The model integrates two key components newly developed to solve compressible flows on unstructured grids with improved accuracy and robustness. A new variant of AUSM scheme, so-called AUSM+-pcp (AUSM+ with pressure-correction projection), has been devised including a pressure-correction projection to the AUSM+ flux splitting, which maintains the exact numerical conservativeness and works well for all Mach numbers. A novel 3th-order, non-oscillatory and less-dissipative reconstruction has been proposed by introducing a multi-dimensional limiting and a BVD (boundary variation diminishing) treatment to the VPM (volume integrated average (VIA) and point value (PV) based multi-moment) reconstruction. The resulting reconstruction scheme, the limited VPM-BVD formulation, is able to resolve both smooth and non-smooth solutions with high fidelity. Benchmark tests have been used to verify the present model. The numerical results substantiate the present model as an accurate and robust unstructured-grid formulation for flows of all Mach numbers.

A tunable finite difference method for fractional differential equations with non-smooth solutions

In this work, a finite difference method of tunable accuracy for fractional differential equations (FDEs) with end-point singularities is developed. Modified weighted shifted Grünwald–Letnikov (WSGL) formulas are proposed to approximate the left and right Riemann–Liouville fractional operators, which show better accuracy than the original WSGL formulas, due to the use of the correction terms. Finite difference schemes are constructed to solve two fractional boundary value problems and a space-fractional Allen–Cahn equation. Even if the singularity of the considered FDEs is unknown, satisfactory numerical solutions can still be obtained by suitably tuning the correction terms. Various numerical examples are presented to verify the effectiveness of the present method, and comparisons with other known methods are also made that demonstrate higher accuracy of the current method.