Discontinuous Solutions Research Articles

Mesh refinement method for solving optimal control problems with nonsmooth solutions using jump function approximations

AbstractA mesh refinement method is described for solving optimal control problems using Legendre‐Gauss‐Radau collocation. The method detects discontinuities in the control solution by employing an edge detection scheme based on jump function approximations. When discontinuities are identified, the mesh is refined with a targeted h‐refinement approach whereby the discontinuity locations are bracketed with mesh points. The remaining smooth portions of the mesh are refined using previously developed techniques. The method is demonstrated on two examples, and results indicate that the method solves optimal control problems with discontinuous control solutions using fewer mesh refinement iterations and less computation time when compared with previously developed methods.

Hybrid high-resolution RBF-ENO method

Essentially nonoscillatory (ENO) and weighted ENO (WENO) methods on equidistant Cartesian grids are widely used to solve partial differential equations with discontinuous solutions. The RBF-ENO method is highly flexible in terms of geometry, but its stencil selection algorithm is computational expensive. In this work, we combine the computationally efficient WENO method and the geometrically flexible RBF-ENO method in a hybrid high-resolution essentially nonoscillatory method to solve hyperbolic conservation laws. The scheme is based on overlapping patches with ghost cells, the RBF-ENO method for unstructured patches and a standard WENO method on structured patches. Furthermore, we introduce a positivity preserving limiter for non-polynomial reconstruction methods to stabilize the hybrid RBF-ENO method for problems with low density or pressure. We show its robustness and flexibility on benchmarks and complex test cases such as the scramjet inflow problem and a conical aerospike nozzle jet simulation.

Discontinuous Galerkin solutions for elliptic problems on polygonal grids using arbitrary-order Bernstein-Bézier functions

In this paper, we present a symmetric interior penalty discontinuous Galerkin finite element discretization for the numerical solution of second-order elliptic partial differential equations on general polygonal meshes using a reduced-space polygonal Bernstein-Bézier functions. They form a space of interpolatory functions (vice local polynomial functions), which satisfy the Lagrange property at their interpolation points. This can enable the use of efficient DGFEM physics-based diffusion preconditioners for the first-order linear transport equation on arbitrary polygons in the computationally-challenging asymptotic diffusion limit. On a polygonal element with n vertices and polynomial degree p, there are n vertex functions, (p−1) functions per edge, and (p−1)(p−2)/2 interior functions that span the {xayb}(a+b)≤p space of bivariate monomials. Numerical experiments highlighting the performance of the proposed method are presented through uniform and adaptive h, p, and hp refinement strategies. Appropriate error rates are observed including hp adaptivity yielding exponential convergence in the presence of singularities in the solution.

Fast Sparse Grid Simulations of Fifth Order WENO Scheme for High Dimensional Hyperbolic PDEs

The weighted essentially non-oscillatory (WENO) schemes, especially the fifth order WENO schemes, are a popular class of high order accurate numerical methods for solving hyperbolic partial differential equations (PDEs). However when the spatial dimensions are high, the number of spatial grid points increases significantly. It leads to large amount of operations and computational costs in the numerical simulations by using nonlinear high order accuracy WENO schemes such as a fifth order WENO scheme. How to achieve fast simulations by high order WENO methods for high spatial dimension hyperbolic PDEs is a challenging and important question. In the literature, sparse-grid technique has been developed as a very efficient approximation tool for high dimensional problems. In a recent work [Lu, Chen and Zhang, Pure and Applied Mathematics Quarterly, 14 (2018) 57-86], a third order finite difference WENO method with sparse-grid combination technique was designed to solve multidimensional hyperbolic equations including both linear advection equations and nonlinear Burgers’ equations. Numerical experiments showed that WENO computations on sparse grids achieved comparable third order accuracy in smooth regions of the solutions and nonlinear stability as that for computations on regular single grids. In application problems, higher than third order WENO schemes are often preferred in order to efficiently resolve the complex solution structures. In this paper, we extend the approach to higher order WENO simulations specifically the fifth order WENO scheme. A fifth order WENO interpolation is applied in the prolongation part of the sparse-grid combination technique to deal with discontinuous solutions. Benchmark problems are first solved to show that significant CPU times are saved while both fifth order accuracy and stability of the WENO scheme are preserved for simulations on sparse grids. The fifth order sparse grid WENO method is then applied to kinetic problems modeled by high dimensional Vlasov based PDEs to further demonstrate large savings of computational costs by comparing with simulations on regular single grids. Several open problems are discussed at last.

Discontinuous solutions of the unsteady boundary-layer equations for a rotating disk of finite radius

Abstract

Arbitrary high‐order extended essentially non‐oscillatory schemes for hyperbolic conservation laws

AbstractAchieving high numerical resolution in smooth regions and robustness near discontinuities within a unified framework is the major concern while developing numerical schemes solving hyperbolic conservation laws, for which the essentially non‐oscillatory (ENO) type scheme is a favorable solution. Therefore, an arbitrary‐high‐order ENO‐type framework is designed in this article. With using a typical five‐point smoothness measurement as the shock‐detector, the present schemes are able to detect discontinuities before spatial reconstructions, and thus more spatial information can be exploited to construct incremental‐width stencils without crossing discontinuities, ensuring ENO property and high‐order accuracy at the same time. The present shock‐detection procedure is specifically examined for justifying its performance of resolving high‐frequency waves, and a standard metric for discontinuous solutions is also applied for measuring the shock‐capturing error of the present schemes, especially regarding the amplitude error in post‐shock regions. In general, the present schemes provide high‐resolution, and more importantly, the schemes are more efficient compared with the typical WENO schemes since only a five‐point smoothness measurement is applied for arbitrary‐high‐order schemes. Numerical results of canonical test cases also provide evidences of the overall performance of the present schemes.

Analysis of Two-Fluid Plasma in the Electromagnetic Hydrodynamics Approximation and Discontinuous Structures in Their Solutions

Various versions of equations of two-fluid plasma called the equations of electromagnetic hydrodynamics are considered. They are a generalization of the classical magnetic hydrodynamics equations obtained by adding dispersive terms. Application of finite-difference methods for solving these equations is analyzed. The Riemann problem is solved numerically, and various types of discontinuity structures that expand with time are considered. These are fast and slow magnetosonic structures and Alfven structures. In the case of moderate amplitudes, fast and slow magnetosonic structures are typical in the theory of nondissipative structures. It is found that due to the disappearance of dispersion, short waves can overturn under certain initial conditions, which requires the analysis of discontinuous solutions or the inclusion of additional dissipative or dispersive terms into the equations. In the case when the fluid dynamic viscosity is added, a shock-type structure is discovered. The evolutionary nature of the discontinuity and the conditions on the discontinuity are studied.

Distributional Profiles for Traveling Waves in the Camassa–Holm Equation

In this paper travelling waves with distributional profiles for the Camassa–Holm equation are studied. Using a product of distributions a new solution concept is introduced which extends the classical one. As a consequence, necessary and sufficient conditions for the propagation of a distributional profile are established and we present examples with discontinuous solutions, measures and even distributions that are not measures. One of these examples may be interpreted as a simple model for a “tsunami” in the setting of shallow water theory. We also prove that, under natural assumptions, profiles belonging to the Sobolev space $$H_{loc} ^{1}({\mathbb {R}})$$ usually considered in the classical weak formulation can be seen as particular cases of our distributional solution concept.

Heteroclinic connections and Dirichlet problems for a nonlocal functional of oscillation type

We consider an energy functional combining the square of the local oscillation of a one-dimensional function with a double-well potential. We establish the existence of minimal heteroclinic solutions connecting the two wells of the potential. This existence result cannot be accomplished by standard methods, due to the lack of compactness properties. In addition, we investigate the main properties of these heteroclinic connections. We show that these minimizers are monotone, and therefore they satisfy a suitable Euler–Lagrange equation. We also prove that, differently from the classical cases arising in ordinary differential equations, in this context the heteroclinic connections are not necessarily smooth, and not even continuous (in fact, they can be piecewise constant). Also, we show that heteroclinics are not necessarily unique up to a translation, which is also in contrast with the classical setting. Furthermore, we investigate the associated Dirichlet problem, studying existence, uniqueness and partial regularity properties, providing explicit solutions in terms of the external data and of the forcing source, and exhibiting an example of discontinuous solution.

The multiplication of distributions in the study of delta shock waves for zero-pressure gasdynamics system with energy conservation laws

In this article, we study the delta shock wave for zero-pressure gasdynamics system with energy conservation laws in the frame of $$\alpha $$ -solutions defined in the setting of distributional products. By reformulating the system, we construct within a convenient space of distributions, all solutions which include discontinuous solutions and Dirac delta measures. We also establish the generalized Rankine–Hugoniot jump conditions for delta shock waves. The $$\alpha $$ -solutions which we constructed coincide with the solution obtained through different methods.

Deep Learning Method Based on Physics Informed Neural Network with Resnet Block for Solving Fluid Flow Problems

Solving fluid dynamics problems mainly rely on experimental methods and numerical simulation. However, in experimental methods it is difficult to simulate the physical problems in reality, and there is also a high-cost to the economy while numerical simulation methods are sensitive about meshing a complicated structure. It is also time-consuming due to the billion degrees of freedom in relevant spatial-temporal flow fields. Therefore, constructing a cost-effective model to settle fluid dynamics problems is of significant meaning. Deep learning (DL) has great abilities to handle strong nonlinearity and high dimensionality that attracts much attention for solving fluid problems. Unfortunately, the proposed surrogate models in DL are almost black-box models and lack interpretation. In this paper, the Physical Informed Neural Network (PINN) combined with Resnet blocks is proposed to solve fluid flows depending on the partial differential equations (i.e., Navier-Stokes equation) which are embedded into the loss function of the deep neural network to drive the model. In addition, the initial conditions and boundary conditions are also considered in the loss function. To validate the performance of the PINN with Resnet blocks, Burger’s equation with a discontinuous solution and Navier-Stokes (N-S) equation with continuous solution are selected. The results show that the PINN with Resnet blocks (Res-PINN) has stronger predictive ability than traditional deep learning methods. In addition, the Res-PINN can predict the whole velocity fields and pressure fields in spatial-temporal fluid flows, the magnitude of the mean square error of the fluid flow reaches to 10−5. The inverse problems of the fluid flows are also well conducted. The errors of the inverse parameters are 0.98% and 3.1% in clean data and 0.99% and 3.1% in noisy data.

Моделирование взаимодействия ударной волны с ограниченным неоднородным слоем газовзвеси гибридным методом крупных частиц

Исследуются задачи взаимодействия ударной волны с ограниченным слоем газовзвеси, внутри которого имеется неоднородность квадратного сечения пониженной или повышенной плотности. Для расчетов используется гибридный метод крупных частиц второго порядка аппроксимации по пространству и времени. Правильность численных разрывных решений, в частности скачков пористости, подтверждается сравнением с асимптотически точными профилями плотности смеси. Приведены аналитические зависимости ослабления ударной волны слоем газовзвеси. Изучены ударно-волновые структуры в двумерных областях и влияние на них релаксационных процессов. The problems of shock wave interaction with a bounded layer of gas suspension is studied in the case when a square-section inhomogeneity of reduced or increased density is situated inside this layer. The hybrid large-particle method of the second-order approximation in space and time is used for calculations. The numerical correctness of discontinuous solutions, in particular jumps of porosity, is confirmed by comparison with the asymptotically exact profiles of the mixture density. Analytical dependences of shock wave attenuation by a gas suspension layer are given. Shock-wave structures in two-dimensional regions and the effect of relaxation processes on them are analyzed.

Anisotropic thermal-conductivity degradation in the phase-field method accounting for crack directionality

Dynamic loads applied to metals may lead to brittle or ductile fracture depending on the imposed strain rates, material properties, and specimen geometry. With an increase in the applied velocity, brittle to ductile failure mode transition may also be observed. Furthermore, at high strain rates, ductile fracture may be preceded by shear bands, which are narrow bands of intense plastic deformation, typically accompanied by a significant rise in temperature. Reliable models are needed to predict the response of metals subject to dynamic loads. In particular, capturing the interplay between heat conduction and crack propagation using the phase-field fracture method is still an open research field. To accurately capture the heat transfer physics across crack surfaces, damage models degrading thermal-conductivity are necessary. While isotropic thermal-conductivity degradation models were proposed, they may lead to errors as they are indifferent to crack directionality. In our current work, a new anisotropic approach is proposed in which thermal-conductivity, which depends on the phase-field gradient, is degraded solely across the crack. It is shown that this approach improves the near-field approximation of temperature and heat flux compared with isotropic degradation, when taking the discontinuous crack solutions as reference. Additionally, the anisotropic degradation approach is implemented in a unified dynamic fracture model and a couple of examples demonstrate the viability of this approach.

An entropy stable high-order discontinuous Galerkin spectral element method for the Baer-Nunziato two-phase flow model

In this work we propose a high-order discretization of the Baer-Nunziato two-phase flow model (Baer and Nunziato (1986) [5]) with closures for interface velocity and pressure adapted to the treatment of discontinuous solutions, and stiffened gas equations of states. We use the discontinuous Galerkin spectral element method (DGSEM), based on collocation of quadrature and interpolation points (Kopriva and Gassner (2010) [47]). The DGSEM uses summation-by-parts (SBP) operators in the numerical quadrature for approximating the integrals over discretization elements (Carpenter et al. (2014) [10]; Gassner et al. (2016) [32]). Here, we build upon the framework provided in (Renac (2019) [55]) for nonconservative hyperbolic systems to modify the integration over cell elements using the SBP operators and replace the physical fluxes with entropy conservative fluctuation fluxes from Castro et al. (2013) [12], while we derive entropy stable numerical fluxes applied at interfaces. This allows to prove a semi-discrete inequality for the cell-averaged physical entropy, while keeping high-order accuracy. The design of the numerical fluxes also formally preserves the kinetic energy at the discrete level. High-order integration in time is performed using strong stability-preserving Runge-Kutta schemes and we propose conditions on the numerical parameters for the positivity of the cell-averaged void fraction and partial densities. The positivity of the cell-averaged solution is extended to nodal values by the use of an a posteriori limiter. The high-order accuracy, nonlinear stability, and robustness of the present scheme are assessed through several numerical experiments in one and two space dimensions.

A nearly-conservative, high-order, forward Lagrange–Galerkin method for the resolution of scalar hyperbolic conservation laws

In this work, we present a novel Lagrange–Galerkin method for the resolution of scalar hyperbolic conservation laws. The scheme considers: (i) a conservative, weak, Lagrangian formulation which is formally discretized in space and in time with arbitrary order of accuracy, (ii) a forward-in-time integration of the fluid trajectories to allow for more stable and efficient time discretizations, (iii) nodal space-discretizations on unstructured triangular meshes and (iv) a novel and simple operator which employs the values of the fine-scale term of the solution to detect and capture the discontinuities. The method has been tested on several benchmark problems –including a hard case of non-convex flux– using a third-order time-integration formula and up to fourth-order finite elements, yielding the expected convergence rates both for smooth and discontinuous solutions. To the best of our knowledge, this is the first Lagrange–Galerkin method for hyperbolic conservation laws in the literature that allows for discontinuous solutions.

The dynamical stress concentration near a cone-shaped crack in a twice-truncated elastic cone

The problem on the dynamical stress concentrations near a cone-shaped crack weakening an elastic twice-truncated cone is solved in case of a steady state torsional oscillations. The problem is considered in a spherical coordinate system due to geometry of cone. The solution is found in a form of superposition of continuous solution and discontinuous one. Continuous solution is found via the use of G. Ya. Popov (2003) integral transform and reduction of the original continuous part of the problem to one-dimensional boundary value problem in vector form in the transform’s domain. The solution then was found in an explicit form. The discontinuous solution was derived with the help of Legendre and variation of Hankel integral transforms. And by solving singular integral equation with the method of orthogonal polynomials the solution of original problem is found. The wave field of an elastic twice truncated cone was investigated. The stress intensity factor near cracks edges was investigated. Also, first eigen frequencies were calculated for different crack’s geometrical characteristics.

Uniformly High-Order Structure-Preserving Discontinuous Galerkin Methods for Euler Equations with Gravitation: Positivity and Well-Balancedness

This paper presents a class of novel high-order accurate discontinuous Galerkin (DG) schemes for the compressible Euler equations under gravitational fields. A notable feature of these schemes is that they are well-balanced for a general hydrostatic equilibrium state, and at the same time, provably preserve the positivity of density and pressure. In order to achieve the well-balanced and positivity-preserving properties simultaneously, a novel DG spatial discretization is carefully designed with suitable source term reformulation and a properly modified Harten-Lax-van Leer contact (HLLC) flux. Based on some technical decompositions as well as several key properties of the admissible states and HLLC flux, rigorous positivity-preserving analyses are carried out. It is proven that the resulting well-balanced DG schemes, coupled with strong stability preserving time discretizations, satisfy a weak positivity property, which implies that one can apply a simple existing limiter to effectively enforce the positivity-preserving property, without losing high-order accuracy and conservation. The proposed methods and analyses are applicable to the Euler system with general equation of state. Extensive one- and two-dimensional numerical tests demonstrate the desired properties of these schemes, including the exact preservation of the equilibrium state, the ability to capture small perturbation of such state, the robustness for solving problems involving low density and/or low pressure, and good resolution for smooth and discontinuous solutions.

Local high-order conservative space-time adaptive mesh refinement with the Runge-Kutta time integration for solving the systems of conservation laws

Numerical simulation is widely used as a main way of researching the processes described by the systems of partial differential equations in the divergence form, like fluid dynamics, traffic flow, wave propagation. To obtain desired accuracy and resolution, the high-order numerical methods are used. In this paper we consider the Runge-Kutta (RK) time discretization, when the space discretization can be high-order finite volume method, discontinuous Galerkin method or others. Presence of discrete analogue of conservation is considered to be very important for such methods as it corresponds to the essential feature of simulated process, becoming especially vital at the discontinuous solutions. The adaptive mesh refinement (AMR) techniques are used to get the numerical solution with more resolution where needed. For the explicit numerical scheme obeying the Courant-Friedrichs-Lewy (CFL) condition, the bigger time integration step can be chosen for the coarser cells. The problem is how to reformulate the numerical scheme near the boundary between cells of different size, keeping the conservation and high-order accuracy. In this paper a new algorithm is suggested. The changes in the numerical method are made only in the vicinity of the boundary. The local character of the developed algorithm allows us to efficiently implement it at modern computer systems. Results of validation tests of developed solver are shown.

Constructing unbounded discontinuous solutions of scalar conservation laws using the Legendre transform

Изучается квазилинейное уравнение первого порядка с нечетной функцией потока, имеющей в нуле единственную точку перегиба. Предложен способ построения разрывных знакопеременных энтропийных решений этого уравнения, основанный на преобразовании Лежандра. Библиография: 18 названий.

Delta waves and vacuum states in the vanishing pressure limit of Riemann solutions to Baer-Nunziato two-phase flow model

<p style='text-indent:20px;'>The phenomena of concentration and cavitation for the Riemann problem of the Baer-Nunziato (BN) two-phase flow model has been investigated in this paper. By using the characteristic analysis method, the formation of <inline-formula><tex-math id="M1">\begin{document}$ \delta- $\end{document}</tex-math></inline-formula>waves and vacuum states are obtained as the pressure for both phases vanish in the BN model. The solid contact wave is carefully dealt. The comparison with the solutions of pressureless two-phase model shows that, two shock waves tend to a <inline-formula><tex-math id="M2">\begin{document}$ \delta- $\end{document}</tex-math></inline-formula>shock solution, and two rarefaction waves tend to a two contact discontinuity solution when the solid contact discontinuity is involved. Moreover, the detailed Riemann solutions for two-phase flow model are given as the double pressure parameters vanish. This may contribute to the design of numerical schemes in the future research.</p>