Abstract: A Newton linearized Galerkin finite element method is proposed to solve nonlinear time fractional parabolic problems with non-smooth solutions in time direction. Iterative processes or corrected schemes become dispensable by the use of the Newton linearized method and graded meshes in the temporal direction. The optimal error estimate in the \(L^2\) -norm is obtained without any time step restrictions dependent on the spatial mesh size. Such unconditional convergence results are proved by including the initial time singularity into concern, while previous unconditional convergent results always require continuity and boundedness of the temporal derivative of the exact solution. Numerical experiments are conducted to confirm the theoretical results. PubDate: 2019-03-22

Abstract: We develop a discretely entropy-stable line-based discontinuous Galerkin method for hyperbolic conservation laws based on a flux differencing technique. By using standard entropy-stable and entropy-conservative numerical flux functions, this method guarantees that the discrete integral of the entropy is non-increasing. This nonlinear entropy stability property is important for the robustness of the method, in particular when applied to problems with discontinuous solutions or when the mesh is under-resolved. This line-based method is significantly less computationally expensive than a standard DG method. Numerical results are shown demonstrating the effectiveness of the method on a variety of test cases, including Burgers’ equation and the Euler equations, in one, two, and three spatial dimensions. PubDate: 2019-03-20

Abstract: We study a pressure-correction ensemble scheme for fast calculation of thermal flow ensembles. The proposed scheme (1) decouples the Boussinesq system into two smaller subphysics problems; (2) decouples the nonlinearity from the incompressibility condition in the Navier–Stokes equations and linearizes the momentum equation so that it reduces to a system of scalar equations; (3) results in linear systems with the same coefficient matrix for all realizations. This reduces the size of linear systems to be solved at each time step and allows efficient direct/iterative linear solvers for fast computation. We prove the scheme is long time stable and first order in time convergent under a time step condition. Numerical tests are provided to confirm the theoretical results and demonstrate the efficiency of the scheme. PubDate: 2019-03-19

Abstract: The recently proposed targeted ENO (TENO) schemes (Fu et al. J Comput Phys 305:333–359, 2016) are demonstrated to feature the controllable low numerical dissipation and sharp shock-capturing property in compressible gas dynamic simulations. However, the application of low-dissipation TENO schemes to ideal magnetohydrodynamics (MHD) is not straightforward. The complex interaction between fluid mechanics and electromagnetism induces extra numerical challenges, including simultaneously preserving the ENO-property, maintaining good numerical robustness and low dissipation as well as controlling divergence errors. In this paper, based on an unstaggered constrained transport framework to control the divergence error, we extend a set of high-order low-dissipation TENO schemes ranging from 5-point to 8-point stencils to solving the ideal MHD equations. A unique set of built-in parameters for each TENO scheme is determined. Different from the TENO schemes in Fu et al. (2016), a modified scale-separation formula is developed. The new formula can achieve stronger scale separation, and it is simpler and more efficient than the previous version as the computation cost of high-order global smoothness measure \({\tau _K}\) is avoided. The performances of tailored schemes are systematically studied by several benchmark simulations. Numerical experiments demonstrate that the TENO schemes in the constrained transport framework are promising to simulate more complex MHD flows. PubDate: 2019-03-18

Abstract: We introduce Hermite-leapfrog methods for first order linear wave systems. The new Hermite-leapfrog methods pair leapfrog time-stepping with the Hermite methods of Goodrich and co-authors et al. (Math Comput 75(254):595–630, 2006). The new schemes stagger field variables in both time and space and are high-order accurate for equations with smooth solutions and coefficients. We provide a detailed description of the method and demonstrate that the method conserves variable quantities. Higher dimensional versions of the method are constructed via tensor products. Numerical evidence and rigorous analysis in one space dimension establish stability and high-order convergence. Experiments demonstrating efficient implementations on a graphics processing unit are also presented. PubDate: 2019-03-18

Abstract: We use optimal control via a distributed exterior field to steer the dynamics of an ensemble of N interacting ferromagnetic particles which are immersed into a heat bath by minimizing a quadratic functional. Using the dynamic programming principle, we show the existence of a unique strong solution of the optimal control problem. By the Hopf–Cole transformation, the associated Hamilton–Jacobi–Bellman equation of the dynamic programming principle may be re-cast into a linear PDE on the manifold \({\mathcal {M}} = ({\mathbb {S}}^{2})^{N}\) , whose classical solution may be represented via Feynman–Kac formula. We use this probabilistic representation for Monte-Carlo simulations to illustrate optimal switching dynamics. PubDate: 2019-03-16

Abstract: We propose an arbitrary-order discontinuous Galerkin method for second-order elliptic problem on general polygonal mesh with only one degree of freedom per element. This is achieved by locally solving a discrete least-squares over a neighboring element patch. Under a geometrical condition on the element patch, we prove an optimal a priori error estimates in the energy norm and in the \(\hbox {L}^2\) norm. The accuracy and the efficiency of the method up to order six on several polygonal meshes are illustrated by a set of benchmark problems. PubDate: 2019-03-14

Abstract: The paper introduces a new algorithm for solving the finite difference equations at the upper time level of an implicit scheme that approximates the Navier–Stokes system in the vorticity-stream function formulation. The algorithm requires no iterations and computes the corresponding discrete solution exactly. It is based on the method of difference potentials and allows one to efficiently address the well-known difficulties typical for this type of formulations—two boundary conditions for the stream function and no boundary conditions for vorticity. for the translation The paper is translated from the Russian by S. Tsynkov (tsynkov@math.ncsu.edu). The original [12] was published more than 20 years ago (the actual Russian citation is [14]). It has not been previously translated into English and went largely unnoticed by the numerical analysis and scientific computing research community. Yet it presents an important contribution to the discipline as it offers a full answer to the question that has long been outstanding. The incompressible Navier–Stokes equations in the vorticity-stream function formulation require two boundary conditions for the stream function and no boundary conditions for vorticity. A standard approach to addressing numerically the apparent overdetermination in one variable and underdetermination in the other was through the use of iterations. Instead, work [12] shows the unambiguous way of discretizing the Navier–Stokes system implicitly and solving the resulting finite difference equations on the upper time level exactly. It is equivalent to deriving the correct non-local boundary condition for vorticity. Note from the translator My Ph.D. advisor Prof. Ryaben’kii and my postdoc mentor Prof. Abarbanel were friends. They interacted closely and participated in joint projects at ICASE (NASA Langley Research Center) in the late 1990s and early 2000s. Prof. Ryaben’kii visited Tel Aviv several times, and Prof. Abarbanel traveled to Moscow in 2013 to celebrate Prof. Ryaben’kii’s 90th birthday. This paper, which was written by Prof. Ryaben’kii with his Ph.D. student at the time, V. Torgashov, provides a well-deserved tribute to a friend from a friend. PubDate: 2019-03-11

Abstract: Fractional differential equations provide a tractable mathematical framework to describe anomalous behavior in complex physical systems, yet they introduce new sensitive model parameters, i.e. derivative orders, in addition to model coefficients. We formulate a sensitivity analysis of fractional models by developing a fractional sensitivity equation method. We obtain the adjoint fractional sensitivity equations, in which we present a fractional operator associated with logarithmic-power law kernel. We further construct a gradient-based optimization algorithm to compute an accurate parameter estimation in fractional model construction. We develop a fast, stable, and convergent Petrov–Galerkin spectral method to numerically solve the coupled system of original fractional model and its corresponding adjoint fractional sensitivity equations. PubDate: 2019-03-09

Abstract: Modeling interfacial dynamics with soluble surfactants in a multiphase system is a challenging task. Here, we consider the numerical approximation of a phase-field surfactant model with fluid flow. The nonlinearly coupled model consists of two Cahn–Hilliard-type equations and incompressible Navier–Stokes equation. With the introduction of two auxiliary variables, the governing system is transformed into an equivalent form, which allows the nonlinear potentials to be treated efficiently and semi-explicitly. By certain subtle explicit-implicit treatments to stress and convective terms, we construct first and second-order time marching schemes, which are extremely efficient and easy-to-implement, for the transformed governing system. At each time step, the schemes involve solving only a sequence of linear elliptic equations, and computations of phase-field variables, velocity and pressure are fully decoupled. We further establish a rigorous proof of unconditional energy stability for the first-order scheme. Numerical results in both two and three dimensions are obtained, which demonstrate that the proposed schemes are accurate, efficient and unconditionally energy stable. Using our schemes, we investigate the effect of surfactants on droplet deformation and collision under a shear flow, where the increase of surfactant concentration can enhance droplet deformation and inhibit droplet coalescence. PubDate: 2019-03-07

Abstract: Parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) are key ingredients in a number of models in physics and financial engineering. In particular, parabolic PDEs and BSDEs are fundamental tools in pricing and hedging models for financial derivatives. The PDEs and BSDEs appearing in such applications are often high-dimensional and nonlinear. Since explicit solutions of such PDEs and BSDEs are typically not available, it is a very active topic of research to solve such PDEs and BSDEs approximately. In the recent article (E et al., Multilevel Picard iterations for solving smooth semilinear parabolic heat equations, arXiv:1607.03295) we proposed a family of approximation methods based on Picard approximations and multilevel Monte Carlo methods and showed under suitable regularity assumptions on the exact solution of a semilinear heat equation that the computational complexity is bounded by \(O( d \, {\varepsilon }^{-(4+\delta )})\) for any \(\delta \in (0,\infty )\) where d is the dimensionality of the problem and \({\varepsilon }\in (0,\infty )\) is the prescribed accuracy. In this paper, we test the applicability of this algorithm on a variety of 100-dimensional nonlinear PDEs that arise in physics and finance by means of numerical simulations presenting approximation accuracy against runtime. The simulation results for many of these 100-dimensional example PDEs are very satisfactory in terms of both accuracy and speed. Moreover, we also provide a review of other approximation methods for nonlinear PDEs and BSDEs from the scientific literature. PubDate: 2019-03-07

Abstract: In this paper, we are concerned with the stochastic Galerkin methods for time-dependent Maxwell’s equations with random input. The generalized polynomial chaos approach is first adopted to convert the original random Maxwell’s equation into a system of deterministic equations for the expansion coefficients (the Galerkin system). It is shown that the stochastic Galerkin approach preserves the energy conservation law. Then, we propose a finite element approach in the physical space to solve the Galerkin system, and error estimates is presented. For the time domain approach, we propose two discrete schemes, namely, the Crank–Nicolson scheme and the leap-frog type scheme. For the Crank–Nicolson scheme, we show the energy preserving property for the fully discrete scheme. While for the classic leap-frog scheme, we present a conditional energy stability property. It is well known that for the stochastic Galerkin approach, the main challenge is how to efficiently solve the coupled Galerkin system. To this end, we design a modified leap-frog type scheme in which one can solve the coupled system in a decouple way—yielding a very efficient numerical approach. Numerical examples are presented to support the theoretical finding. PubDate: 2019-03-07

Abstract: This article is concerned with the mathematical and numerical analysis of a steady phase change problem for non-isothermal incompressible viscous flow. The system is formulated in terms of pseudostress, strain rate and velocity for the Navier–Stokes–Brinkman equation, whereas temperature, normal heat flux on the boundary, and an auxiliary unknown are introduced for the energy conservation equation. In addition, and as one of the novelties of our approach, the symmetry of the pseudostress is imposed in an ultra-weak sense, thanks to which the usual introduction of the vorticity as an additional unknown is no longer needed. Then, for the mathematical analysis two variational formulations are proposed, namely mixed-primal and fully-mixed approaches, and the solvability of the resulting coupled formulations is established by combining fixed-point arguments, Sobolev embedding theorems and certain regularity assumptions. We then construct corresponding Galerkin discretizations based on adequate finite element spaces, and derive optimal a priori error estimates. Finally, numerical experiments in 2D and 3D illustrate the interest of this scheme and validate the theory. PubDate: 2019-03-06

Abstract: This work examines the development of an entropy conservative (for smooth solutions) or entropy stable (for discontinuous solutions) space–time discontinuous Galerkin (DG) method for systems of nonlinear hyperbolic conservation laws. The resulting numerical scheme is fully discrete and provides a bound on the mathematical entropy at any time according to its initial condition and boundary conditions. The crux of the method is that discrete derivative approximations in space and time are summation-by-parts (SBP) operators. This allows the discrete method to mimic results from the continuous entropy analysis and ensures that the complete numerical scheme obeys the second law of thermodynamics. Importantly, the novel method described herein does not assume any exactness of quadrature in the variational forms that naturally arise in the context of DG methods. Typically, the development of entropy stable schemes is done on the semidiscrete level ignoring the temporal dependence. In this work, we demonstrate that creating an entropy stable DG method in time is similar to the spatial discrete entropy analysis, but there are important (and subtle) differences. Therefore, we highlight the temporal entropy analysis throughout this work. For the compressible Euler equations, the preservation of kinetic energy is of interest besides entropy stability. The construction of kinetic energy preserving (KEP) schemes is, again, typically done on the semidiscrete level similar to the construction of entropy stable schemes. We present a generalization of the KEP condition from Jameson to the space–time framework and provide the temporal components for both entropy stability and kinetic energy preservation. The properties of the space–time DG method derived herein are validated through numerical tests for the compressible Euler equations. Additionally, we provide, in appendices, how to construct the temporal entropy stable components for the shallow water or ideal magnetohydrodynamic (MHD) equations. PubDate: 2019-03-04

Abstract: A novel second-order numerical approximation for the Riemann–Liouville tempered fractional derivative, called the tempered fractional-compact difference formula is derived by using the tempered Grünwald difference operator and its asymptotic expansion. Using the relationship between Riemann–Liouville and the Caputo tempered fractional derivatives, then the constructed approximation formula is applied to deal with the time-Caputo-tempered partial differential equation in time, while the spatial Riesz derivative are discretized by the fourth-order compact numerical differential formulas. By using the energy method, it is proved that the proposed algorithm to be unconditionally stable and convergent with order \({\mathcal {O}}\left( \tau ^2+h_1^4+h_2^4\right) \) , where \(\tau \) is the temporal stepsize and \(h_1,h_2\) are the spatial stepsizes respectively. Finally, some numerical examples are performed to testify the effectiveness of the obtained algorithm. PubDate: 2019-03-02

Abstract: This paper is devoted to virtual element methods for solving elliptic variational inequalities (EVIs) of the second kind. First, a general framework is provided for the numerical solution of the EVIs and for its error analysis. Then virtual element methods are applied to solve two representative EVIs: a simplified friction problem and a frictional contact problem. Optimal order error estimates are derived for the virtual element solutions of the two representative EVIs, including the effects of numerical integration for the non-smooth term in the EVIs. A fast solver is introduced to solve the discrete problems. Several numerical examples are included to show the numerical performance of the proposed methods. PubDate: 2019-03-01

Abstract: The subject of the present paper is to apply the Jacobi spectral collocation method for multidimensional linear Volterra integral equation with a weakly singular kernel. Here, we assume that the solution is sufficiently smooth. An error analysis has been provided which justifies that the approximate solution converges exponentially to the exact solution. Finally, two numerical examples are given to clarify the efficiency and accuracy of the method. PubDate: 2019-02-25

Abstract: In this paper, we study a shape optimization problem in two dimensions where the objective function is the convex combination of two sequential Steklov eigenvalues of a domain with a fixed area constraint. We show the existence of the optimal domain and the nondecreasing, Lipschitz continuity, and convexity of the optimal objective function with respect to the convex combination constant. On one-parameter family of rectangular domains, asymptotic behaviors of lower eigenvalues are found. For general shapes, numerical approaches are used to find optimal shapes. The range of the first two Steklov eigenvalues are discussed for several one-parameter families of shapes including Cassini oval shapes and Hippopede shapes. PubDate: 2019-02-23

Abstract: In this work, we propose efficient and accurate numerical algorithms based on difference potentials method for numerical solution of chemotaxis systems and related models in 3D. The developed algorithms handle 3D irregular geometry with the use of only Cartesian meshes and employ Fast Poisson Solvers. In addition, to further enhance computational efficiency of the methods, we design a difference-potentials-based domain decomposition approach which allows mesh adaptivity and easy parallelization of the algorithm in space. Extensive numerical experiments are presented to illustrate the accuracy, efficiency and robustness of the developed numerical algorithms. PubDate: 2019-02-20

Abstract: Tensor representations allow compact storage and efficient manipulation of multi-dimensional data. Based on these, tensor methods build low-rank subspaces for the solution of multi-dimensional and multi-parametric models. However, tensor methods cannot always be implemented efficiently, specially when dealing with non-linear models. In this paper, we discuss the importance of achieving a tensor representation of the model itself for the efficiency of tensor-based algorithms. We investigate the adequacy of interpolation rather than projection-based approaches as a means to enforce such tensor representation, and propose the use of cross approximations for models in moderate dimension. Finally, linearization of tensor problems is analyzed and several strategies for the tensor subspace construction are proposed. PubDate: 2019-02-19