Mathematics

In this work we present an application of modern deep learning methodologies to the numerical solution of partial differential equations in transport models. More specifically, we employ a supervised deep neural network that takes into account the equation and initial conditions of the model. We apply it to the Riemann problems over the inviscid nonlinear Burger's equation, whose solutions might develop discontinuity (shock wave) and rarefaction, as well as to the classical one-dimensional Buckley-Leverett two-phase problem. The Buckley-Leverett case is slightly more complex and interesting because it has a non-convex flux function with one inflection point. Our results suggest that a relatively simple deep learning model was capable of achieving promising results in such challenging tasks, providing numerical approximation of entropy solutions with very good precision and consistent to classical as well as to recently novel numerical methods in these particular scenarios.

In the literature, there exist several studies on symbol-based multigrid methods for the solution of linear systems having structured coefficient matrices. In particular, the convergence analysis for such methods has been obtained in an elegant form in the case of Toeplitz matrices generated by a scalar-valued function. In the block-Toeplitz setting, that is, in the case where the matrix entries are small generic matrices instead of scalars, some algorithms have already been proposed regarding specific applications and a first rigorous convergence analysis has been performed in [7]. However, with the existent symbol-based theoretical tools, it is still not possible to prove the convergence of many multigrid methods known in the literature. This paper aims to generalize the previous results giving more general sufficient conditions on the symbol of the grid transfer this http URL particular, we treat matrix-valued trigonometric polynomials which can be non-diagonalizable and singular at all points and we express the new conditions in terms of the eigenvectors associated with the ill-conditioned subspace. Moreover, we extend the analysis to the V-cycle method proving a linear convergence rate under stronger conditions, which resemble those given in the scalar case. In order to validate our theoretical findings, we present a classical block structured problem stemming from a FEM approximation of a second order differential problem. We focus on two multigrid strategies that use the geometric and the standard bisection grid transfer operators and we prove that both fall into the category of projectors satisfying the proposed conditions. In addition, using a tensor product argument, we provide a strategy to construct efficient V-cycle procedures in the block multilevel setting.

We introduce a symmetric fractional-order reduction (SFOR) method to construct numerical algorithms on general nonuniform temporal meshes for semilinear fractional diffusion-wave equations. By using the novel order reduction method, the governing problem is transformed to an equivalent coupled system, where the explicit orders of time-fractional derivatives involved are all α/2 (1<α<2) . The linearized L1 scheme and Alikhanov scheme are then proposed on general time meshes. Under some reasonable regularity assumptions and weak restrictions on meshes, the optimal convergence is derived for the two kinds of difference schemes by H 2 energy method. An adaptive time stepping strategy which based on the (fast linearized) L1 and Alikhanov algorithms is designed for the semilinear diffusion-wave equations. Numerical examples are provided to confirm the accuracy and efficiency of proposed algorithms.

This paper describes a systematic method of numerically computing and indexing fixed points of z z w for fixed z or equivalently, the roots of T 2 (w;z)=w− z z w . The roots are computed using a modified version of fixed-point iteration and indexed by integer triplets {n,m,p} which associate a root to a unique branch of T 2 . This naming convention is proposed sufficient to enumerate all roots of the function with (n,m) enumerated by Z 2 . However, branches near the origin can have multiple roots. These cases are identified by the third parameter p . This work was done with rational or symbolic values of z enabling arbitrary precision arithmetic. A selection of roots up to order { 10 12 , 10 12 ,p} with |z|≤ 10 12 was used as test cases. Results were accurate to the precision used in the computations, generally between 30 and 100 digits. Mathematica ver. 12 was used to implement the algorithms.

A finite difference algorithm based on the integral Laguerre transform in time for solving a three-dimensional one-way wave equation is proposed. This allows achieving high accuracy of calculation results. In contrast to the Fourier method, the approach does not need to solve systems of linear algebraic equations with indefinite matrices. To filter the unstable components of a wave field, Richardson extrapolation or spline approximation can be used. However, these methods impose additional limitations on the integration step in depth. This problem can be solved if the filtering is performed not in the direction of extrapolation of the wave field, but in a horizontal plane. This approach called for fast methods of converting the Laguerre series coefficients into the Fourier series coefficients and vice versa. The high stability of the new algorithm allows calculations with a large depth step without loss of accuracy and, in combination with Marchuk-Strang splitting, this can significantly reduce the calculation time. Computational experiments are performed. The results have shown that this algorithm is highly accurate and efficient in solving the problems of seismic migration.

We present a 3D hybrid method which combines the Finite Element Method (FEM) and the Spectral Boundary Integral method (SBIM) to model nonlinear problems in unbounded domains. The flexibility of FEM is used to model the complex, heterogeneous, and nonlinear part -- such as the dynamic rupture along a fault with near fault plasticity -- and the high accuracy and computational efficiency of SBIM is used to simulate the exterior half spaces perfectly truncating all incident waves. The exact truncation allows us to greatly reduce the domain of spatial discretization compared to a traditional FEM approach, leading to considerable savings in computational cost and memory requirements. The coupling of FEM and SBIM is achieved by the exchange of traction and displacement boundary conditions at the computationally defined boundary. The method is suited to implementation on massively parallel computers. We validate the developed method by means of a benchmark problem. Three more complex examples with a low velocity fault zone, low velocity off-fault inclusion, and interaction of multiple faults, respectively, demonstrate the capability of the hybrid scheme in solving problems of very large sizes. Finally, we discuss potential applications of the hybrid method for problems in geophysics and engineering.

Isogeometric Analysis (IGA) is a computational technique for the numerical approximation of partial differential equations (PDEs). This technique is based on the use of spline-type basis functions, that are able to hold a global smoothness and allow to exactly capture a wide set of common geometries. The current rise of this approach has encouraged the search of fast solvers for isogeometric discretizations and nowadays this topic is full of interest. In this framework, a desired property of the solvers is the robustness with respect to both the polinomial degree p and the mesh size h . For this task, in this paper we propose a two-level method such that a discretization of order p is considered in the first level whereas the second level consists of a linear or quadratic discretization. On the first level, we suggest to apply one single iteration of a multiplicative Schwarz method. The choice of the block-size of such an iteration depends on the spline degree p , and is supported by a local Fourier analysis (LFA). At the second level one is free to apply any given strategy to solve the problem exactly. However, it is also possible to get an approximation of the solution at this level by using an h− multigrid method. The resulting solver is efficient and robust with respect to the spline degree p . Finally, some numerical experiments are given in order to demonstrate the good performance of the proposed solver.

In this work, we propose a state redistribution method for high order discontinuous Galerkin methods on curvilinear embedded boundary grids. State redistribution relaxes the overly restrictive CFL condition that results from arbitrarily small cut cells when explicit time steppers are used. Thus, the scheme can take take time steps that are proportional to the size of cells in the background grid. The discontinuous Galerkin scheme is stabilized by postprocessing the numerical solution after each stage or step of an explicit time stepping method. This is done by temporarily coarsening, or merging, the small cells into larger, possibly overlapping neighborhoods using a special weighted inner product. Then, the numerical solution on the neighborhoods is refined back onto the base grid in a conservative fashion. The advantage of this approach is that it uses only basic mesh information that is already available in many cut cell codes and does not require complex geometric manipulations. We prove that state redistribution is conservative and p -exact. Finally, we solve a number of test problems that demonstrate the encouraging potential of this technique for applications on curvilinear embedded geometries. Numerical experiments reveal that our scheme converges with order p+1 in L 1 and between p and p+1 in L ??.

In this paper we consider a class of boundary value problems for third order nonlinear functional differential equation. By the reduction of the problem to operator equation we establish the existence and uniqueness of solution and construct a numerical method for solving it. We prove that the method is of second order accuracy and obtain an estimate for total error. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the numerical method. The approach used for the third order nonlinear functional differential equation can be applied to functional differential equations of any orders.

We propose a unifying framework for the matrix-based formulation and analysis of discontinuous Galerkin (DG) and flux reconstruction (FR) methods for conservation laws on general unstructured grids. Within such an algebraic framework, the multidimensional summation-by-parts (SBP) property is used to establish the discrete equivalence of strong and weak formulations, as well as the conservation and energy stability properties of a broad class of DG and FR schemes. Specifically, the analysis enables the extension of the equivalence between the strong and weak forms of the discontinuous Galerkin collocation spectral-element method demonstrated by Kopriva and Gassner (J Sci Comput 44:136-155, 2010) to more general nodal and modal DG formulations, as well as to the Vincent-Castonguay-Jameson-Huynh (VCJH) family of FR methods. Moreover, new algebraic proofs of conservation and energy stability for DG and VCJH schemes with respect to suitable quadrature rules and discrete norms are presented, in which the SBP property serves as a unifying mechanism for establishing such results. Numerical experiments are provided for the two-dimensional linear advection and Euler equations, highlighting the design choices afforded for methods within the proposed framework and corroborating the theoretical analysis.