Local Asymptotic Expansion Research Articles

The late to early time behaviour of an expanding plasma: hydrodynamisation from exponential asymptotics

We use exponential asymptotics to match the late time temperature evolution of an expanding conformally invariant fluid to its early time behaviour. We show that the rich divergent transseries asymptotics at late times can be used to interpolate between the two regimes with exponential accuracy using the well-established methods of hyperasymptotics, Borel resummation and transasymptotics. This approach is generic and can be applied to any interpolation problem involving a local asymptotic transseries expansion as well as knowledge of the solution in a second region away from the expansion point. Moreover, we present global analytical properties of the solutions such as analytic approximations to the locations of the square-root branch points, exemplifying how the summed transseries contains within itself information about the observable in regions with different asymptotics.

A mathematical model for a steady-state seepage flow of groundwater under a reinforced concrete dam

A numerical technique is applied to solve a biharmonic equation problem of a steady-state seepage flow of water through a thin layer of soil, under an impermeable dam. Due to the existence of a point boundary singularity and particular boundary conditions in the problem of the present study, the biharmonic equation is selected to be the governing equation. The potential function of the problem is approximated by the leading terms of the local asymptotic solution expansion. These terms are also used to weight the governing biharmonic equation in the Galerkin sense. Implementing Green’s theorem twice, the discretized equations are reduced to boundary integrals. The system of linear equations is completed by weakly enforcing the Dirichlet boundary conditions by means of Lagrange multipliers. The values of the latter are calculated together with the singular coefficients. The numerical results obtained with the method, indicate high accuracy and very fast convergence with the number of singular functions and the number of Lagrange multipliers. The volume of water per unit time q flowing through the soil in the horizontal direction, under the dam, which is directly proportional to the directional derivative of the potential function, is approximated by the derivatives of the leading terms of the local asymptotic solution expansion. Values of q and the corresponding discharge velocity, obtained with the method, compare favourably with the values given by FEM and soil Mechanics theory. The favourable numerical behaviour of this numerical technique, which in the bibliography is known as singular function boundary integral method (SFBIM), can be exploited in numerous ways. Combining the SFBIM with standard schemes, such as the FEM, appears to be an interesting extension of the method for future work, aiming at enforcing the strengths of the SFBIM in problems with boundary singularities.

A local asymptotic expansion for a solution of the Stokes system

We consider solutions of the Stokes system in a neighborhood of apoint in which the velocity $u$ vanishes of order $d$. We prove that thereexists a divergence-free polynomial $P$ in $x$ with $t$-dependent coefficientsof degree $d$ which approximates the solution $u$ of order $d+\alpha$for certain $\alpha>0$.The polynomial $P$ satisfies a Stokes equation with a forcing term whichis a sum of two polynomials in $x$ of degrees $d-1$ and $d$.

A Novel Solver for the Generalized Riemann Problem Based on a Simplified LeFloch–Raviart Expansion and a Local Space–Time Discontinuous Galerkin Formulation

In a wide class of high order shock-capturing methods for hyperbolic conservation laws, the solution of the conservation law is represented at each time-step by a piecewise smooth function (say, a polynomial reconstructed from cell-averages or an approximation in a finite element space). To maintain a sharp resolution of shock waves, jumps at the cell boundaries are allowed. The resulting initial value problem with piecewise smooth but discontinuous initial data is called the generalized Riemann problem. We present a new solver for the generalized Riemann problem based on a simplified version of a local asymptotic series expansion constructed by LeFloch and Raviart (Ann Inst H Poincare Anal Non Lineaire 5:179---207, 1988). Contrary to the original approach, in our new solver no higher order flux derivatives and other nonlinear terms need to be computed. Moreover, we introduce a new variant of the local space---time DG method of Dumbser et al. (J Comput Phys 227:3971---4001, 2008), that allows us to use a direct solution strategy for the generalized Riemann problem without relying on a Cauchy---Kovalevskaya procedure for the flux computation.

The Singular Function Boundary Integral Method for the 2-D and 3-D Laplace Equation Problems in Mechanics, with a Boundary Singularity

In this study the Singular Function Boundary Integral Method (SFBIM) is implemented in the case of a planar elliptic boundary value problem in Mechanics, with a point boundary singularity. The method is also extended in the case of a typical problem of Solid Mechanics, concerning the Laplace equation problem in three dimensions, defined in a domain with a straight edge singularity on the surface boundary. In both the 2-D and 3-D cases, the general solution of the Laplace equation is approximated by the leading terms (which contain the singular functions) of the local asymptotic solution expansion. The singular functions are used to weight the governing equation in the Galerkin sense. For the 2-D Laplacian model problem of this study, which is defined over a domain with a re-entrant corner, the resulting discretized equations are reduced to boundary integrals by means of Green’s second identity. For the 3-D model problem of this work, the volume integrals of the discretized equations are reduced to surface integrals by implementing Gauss’ divergence theorem. The Dirichlet boundary conditions are then weakly enforced by means of Lagrange multipliers. The values of the latter are calculated together with the singular coefficients, in the 2-D case or the Edge Flux Intensity Functions (EFIFs), in the 3-D model problem, which appear in the local solution expansion. For the planar problem, the numerical results are favorably compared with the analytic solution. Especially for the extension of the method in three dimensions, the preliminary numerical results compare favorably with available post-processed finite element results.

Recovering a stochastic process from super-resolution noisy ensembles of single-particle trajectories

Recovering a stochastic process from noisy ensembles of single-particle trajectories is resolved here using the coarse-grained Langevin equation as a model. The massive redundancy contained in single-particle tracking data allows recovering local parameters of the underlying physical model. We use several parametric and nonparametric estimators to compute the first and second moments of the process, to recover the local drift, its derivative, and the diffusion tensor, and to deconvolve the instrumental from the physical noise. We use numerical simulations to also explore the range of validity for these estimators. The present analysis allows defining what can exactly be recovered from statistics of super-resolution microscopy trajectories used for characterizing molecular trafficking underlying cellular functions.

Approximate solutions of generalized Riemann problems for nonlinear systems of hyperbolic conservation laws

We study analytical properties of the Toro-Titarev solver for generalized Riemann problems (GRPs), which is the heart of the flux computation in ADER generalized Godunov schemes. In particular, we compare the Toro-Titarev solver with a local asymptotic expansion developed by LeFloch and Raviart. We show that for scalar problems the Toro-Titarev solver reproduces the truncated Taylor series expansion of LeFloch-Raviart exactly, whereas for nonlinear systems the Toro-Titarev solver introduces an error whose size depends on the height of the jump in the initial data. Thereby, our analysis answers open questions concerning the justification of simplifying steps in the Toro-Titarev solver. We illustrate our results by giving the full analysis for a nonlinear 2-by-2 system and numerical results for shallow water equations.

The singular function boundary integral method for an elastic plane stress wedge beam problem with a point boundary singularity

The singular function boundary integral method (SFBIM) is applied for the numerical solution of a 2-D Laplace model problem of a perfectly elastic wedge beam under plane stress conditions. The beam has a point boundary singularity, it includes a curved boundary part and is subjected to non-trivial distributed external loading. The implemented solution method converges for this special model problem extremely fast. The numerical estimates attained for the leading singular coefficients of the local asymptotic expansion and the stress and strain fields are highly accurate, as verified by comparison with the available analytical solution.

The singular function boundary integral method for 3-D Laplacian problems with a boundary straight edge singularity

Three-dimensional Laplace problems with a boundary straight-edge singularity caused by two intersecting flat planes are considered. The solution in the neighbourhood of the straight edge can be expressed as an asymptotic expansion involving the eigenpairs of the analogous two-dimensional problem in polar coordinates, which have as coefficients the so-called edge flux intensity functions (EFIFs). The EFIFs are functions of the axial coordinate, the higher derivatives of which appear in an inner infinite series in the expansion. The objective of this work is to extend the singular function boundary integral method (SFBIM), developed for two-dimensional elliptic problems with point boundary singularities [G.C. Georgiou, L. Olson, G. Smyrlis, A singular function boundary integral method for the Laplace equation, Commun. Numer. Methods Eng. 12 (1996) 127–134] for solving the above problem and directly extracting the EFIFs. Approximating the latter by either piecewise constant or linear polynomials eliminates the inner infinite series in the local expansion and allows the straightforward extension of the SFBIM. As in the case of two-dimensional problems, the solution is approximated by the leading terms of the local asymptotic solution expansion. These terms are also used to weight the governing harmonic equation in the Galerkin sense. The resulting discretized equations are reduced to boundary integrals by means of the divergence theorem. The Dirichlet boundary conditions are then weakly enforced by means of Lagrange multipliers. The values of the latter are calculated together with the coefficients of the EFIFs. The SFBIM is applied to a test problem exhibiting fast convergence of order k+1 (k being the order of the approximation of the EFIFs) in the L2-norm and leading to accurate estimates for the EFIFs.

A partition of unity FEM for time‐dependent diffusion problems using multiple enrichment functions

SUMMARYAn enriched partition of unity FEM is developed to solve time‐dependent diffusion problems. In the present formulation, multiple exponential functions describing the spatial and temporal diffusion decay are embedded in the finite element approximation space. The resulting enrichment is in the form of a local asymptotic expansion. Unlike previous works in this area where the enrichment must be updated at each time step, here, the temporal decay in the solution is embedded in the asymptotic expansion. Thus, the system matrix that is evaluated at the first time step may be decomposed and retained for every next time step by just updating the right‐hand side of the linear system of equations. The advantage is a significant saving in the computational effort where, previously, the linear system must be reevaluated and resolved at every time step. In comparison with the traditional finite element analysis with p‐version refinements, the present approach is much simpler, more efficient, and yields more accurate solutions for a prescribed number of DoFs. Numerical results are presented for a transient diffusion equation with known analytical solution. The performance of the method is analyzed on two applications: the transient heat equation with a single source and the transient heat equation with multiple sources. The aim of such a method compared with the classical FEM is to solve time‐dependent diffusion applications efficiently and with an appropriate level of accuracy. Copyright © 2012 John Wiley & Sons, Ltd.

Analysis of the singular function boundary integral method for a biharmonic problem with one boundary singularity

AbstractIn this article, we analyze the singular function boundary integral method (SFBIM) for a two‐dimensional biharmonic problem with one boundary singularity, as a model for the Newtonian stick‐slip flow problem. In the SFBIM, the leading terms of the local asymptotic solution expansion near the singular point are used to approximate the solution, and the Dirichlet boundary conditions are weakly enforced by means of Lagrange multiplier functions. By means of Green's theorem, the resulting discretized equations are posed and solved on the boundary of the domain, away from the point where the singularity arises. We analyze the convergence of the method and prove that the coefficients in the local asymptotic expansion, also referred to as stress intensity factors, are approximated at an exponential rate as the number of the employed expansion terms is increased. Our theoretical results are illustrated through a numerical experiment. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011

The Singular Function Boundary Integral Method for singular Laplacian problems over circular sections

The Singular Function Boundary Integral Method (SFBIM) for solving two-dimensional elliptic problems with boundary singularities is revisited. In this method the solution is approximated by the leading terms of the asymptotic expansion of the local solution, which are also used to weight the governing partial differential equation. The singular coefficients, i.e., the coefficients of the local asymptotic expansion, are thus primary unknowns. By means of the divergence theorem, the discretized equations are reduced to boundary integrals and integration is needed only far from the singularity. The Dirichlet boundary conditions are then weakly enforced by means of Lagrange multipliers, the discrete values of which are additional unknowns. In the case of two-dimensional Laplacian problems, the SFBIM converges exponentially with respect to the numbers of singular functions and Lagrange multipliers. In the present work the method is applied to Laplacian test problems over circular sectors, the analytical solution of which is known. The convergence of the method is studied for various values of the order p of the polynomial approximation of the Lagrange multipliers (i.e., constant, linear, quadratic, and cubic), and the exact approximation errors are calculated. These are compared to the theoretical results provided in the literature and their agreement is demonstrated.

The singular function boundary integral method for biharmonic problems with crack singularities

We use the singular function boundary integral method (SFBIM) to solve two model fracture problems on the plane. In the SFBIM, the solution is approximated by the leading terms of the local asymptotic solution expansion, which are also used to weight the governing biharmonic equation in the Galerkin sense. The discretized equations are reduced to boundary integrals by means of the divergence theorem and the Dirichlet boundary conditions are weakly enforced by means of Lagrange multipliers. The main advantage of the method is that the leading stress intensity factors (SIFs) are calculated directly together with the Lagrange multipliers, i.e. no post-processing of the numerical solution is necessary. The numerical results for the two model problems show the fast convergence of the method and compare well with those of the collocation Trefftz method.

Solution of the planar Newtonian stick–slip problem with the singular function boundary integral method

AbstractA singular function boundary integral method (SFBIM) is proposed for solving biharmonic problems with boundary singularities. The method is applied to the Newtonian stick–slip flow problem. The streamfunction is approximated by the leading terms of the local asymptotic solution expansion which are also used to weight the governing biharmonic equation in the Galerkin sense. By means of the divergence theorem the discretized equations are reduced to boundary integrals. The Dirichlet boundary conditions are weakly enforced by means of Lagrange multipliers, the values of which are calculated together with the singular coefficients. The method converges very fast with the number of singular functions and the number of Lagrange multipliers, and accurate estimates of the leading singular coefficients are obtained. Comparisons with the analytical solution and results obtained with other numerical methods are also made. Copyright © 2005 John Wiley & Sons, Ltd.

Solving Laplacian problems with boundary singularities: a comparison of a singular function boundary integral method with the p/ hp version of the finite element method

We solve a Laplacian problem over an L-shaped domain using a singular function boundary integral method as well as the p/ hp finite element method. In the former method, the solution is approximated by the leading terms of the local asymptotic solution expansion, and the unknown singular coefficients are calculated directly. In the latter method, these coefficients are computed by post-processing the finite element solution. The predictions of the two methods are discussed and compared with recent numerical results in the literature.

Enriched partition-of-unity finite element method for stress intensity factors at crack tips

The traditional finite element method (FEM) is very versatile but is not efficient in dealing with crack-tip problems, even within the linear elastic regime. It is due to the presence of singular stress field in the vicinity of a crack tip. Although many endeavors have been reported, no efficient and ‘one-for-all’ type of solution has emerged yet. This paper presents a unified approach, which is particularly efficient in solving ‘stress intensity factor’ at crack tips. The present approach is based on the partition-of-unity (PU) method, which is known as one of the meshless methods. Actually, the PU method encompasses the traditional FEM. In the present PU formulation, an analytical solution exactly describing the stress field around the crack tip is embedded in the FE shape function. It results in an enriched approximation, which is mathematically singular and in the form of local asymptotic expansion. It yields more accurate solutions. In comparison with the traditional FE analysis with p-version refinements, the present approach is much simpler, more robust and efficient. On the other hand, the notorious difficulty encountered in the meshless method in satisfying the essential boundary conditions is circumvented. Moreover, the present formulation has another merit that the stress intensity factors at crack tip can be obtained directly from numerical solution without any post-processing. This paper includes illustrative examples showing the rate of convergence and efficiency, benchmarked against other available solutions, in particular those obtained by p-version FEM. The efficiency and versatility of present formulation are demonstrated.

The solution of Laplacian problems over L‐shaped domains with a singular function boundary integral method

AbstractThe singular function boundary integral method is applied for the solution of a Laplace equation problem over an L‐shaped domain. The solution is approximated by the leading terms of the local asymptotic solution expansion, while the Dirichlet boundary conditions are weakly enforced by means of Lagrange multipliers. Estimates of great accuracy are obtained for the leading singular coefficients, as well as for the Lagrange multipliers. Comparisons are made with recent numerical results in the literature. Copyright © 2002 John Wiley & Sons, Ltd.

The design and analysis of the Generalized Finite Element Method

In this paper, we introduce the Generalized Finite Element Method (GFEM) as a combination of the classical Finite Element Method (FEM) and the Partition of Unity Method (PUM). The standard finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data; e.g., the singular functions obtained from the local asymptotic expansion of the exact solution in the neighborhood of a corner point, etc. The special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape-functions and pasted together with the existing finite element basis to construct an augmented conforming finite element space. In this way, the local approximability afforded by the special functions is included in the approximation, while maintaining the existing infrastructure of finite element codes. The major features of the GFEM are: (1) the essential boundary conditions can be imposed exactly as in the standard FEM, unlike other partition of unity based methods where this is a major issue; (2) the accuracy of the numerical integration of the entries of the stiffness matrix and load vector is controlled adaptively so that the errors in integration of the special functions do not affect the accuracy of the constructed approximation (this issue also has not been sufficiently addressed in other implementations of partition of unity or meshless methods); and (3) linear dependencies in the system of equations are resolved by employing an easy modification of the direct linear solver. The power of the GFEM for solving problems in domains with complex geometry with less error and less computer resources than the standard FEM is illustrated by numerical examples.

A singular function boundary integral method for the Laplace equation

The authors present a new singular function boundary integral method for the numerical solution of problems with singularities which is based on approximation of the solution by the leading terms of the local asymptotic expansion. The essential boundary conditions are weakly enforced by means of appropriate Lagrange multipliers. The method is applied to a benchmark Laplace-equation problem, the Motz problem, giving extremely accurate estimates for the leading singular coefficients. The method converges exponentially with the number of singular functions and requires a low computational cost. Comparisons are made to the analytical solution and other numerical methods.

Quantum gravitational effects in an isotropic Universe with torsion

In the framework of the quasiclassical self-consistent approach we consider a nonminimally coupled scalar field which serves as a source of torsion in an isotropic homogeneous Universe. We obtain a local asymptotic expansion for the propagator of the scalar field and the effective action, taking into account the effects of vacuum polarization and interaction of scalar particles with the self-consistent gravitational field. The leading polarization contributions can be represented in general-covariant form. A simplified cosmological model is constructed which takes into account the interaction.