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Cell migration is fundamental in a wide variety of physiological and pathological phenomena, being exploited in biomedical engineering as well. In this respect, we here present a hybrid non-local integro-differential model where a representative cell, reproduced by a point particle with an orientation, moves on a planar domain upon signals coming from environmental variables. From a numerical point of view, non-locality implies the need to evaluate integral terms which may present non-regular integrand functions because of heterogeneities in the environmental conditions and/or in cell sensing region. Having in mind multicellular applications, we here propose a robust computational method able to handle such non-regularities. The procedure is based on low order Runge–Kutta methods and on an ad hoc application of the Gauss–Legendre quadrature rule. The accuracy and efficiency of the resulting computational method is then tested by selected benchmark settings. In this context, the ad hoc application of the quadrature rule reveals to be crucial to obtain a high accuracy with a remarkably low number of quadrature nodes with respect to the standard Gauss–Legendre quadrature formula, and which thus results in a reduced overall computational cost. Finally, the proposed method is further coupled with the cubic spline interpolation scheme which allows to deal also with possible poor (i.e., point-wise defined) molecular spatial information. The performed simulations (which accounts also for different scenarios) show how the interpolation of the molecular variables affects the efficiency of the overall method and further justify the proposed procedure.

We present a new numerical approach to solve 2D exterior Helmholtz problems defined in unbounded domains. This consists in reducing the infinite region to a finite computational one Ω , by the introduction of an artificial boundary B , and by applying in Ω a Virtual Element Method (VEM). The latter is coupled with a Boundary Integral Non Reflecting Condition defined on B (in short BI-NRBC), discretized by a standard collocation Boundary Element Method (BEM). We show that, by choosing the same approximation order of the VEM and of the BI-NRBC discretization spaces, the corresponding method allows to obtain the optimal order of convergence. We test the efficiency and accuracy of the proposed approach on various numerical examples, arising both from literature and real life application problems.

We consider (transient) 3D elastic wave propagation problems in unbounded isotropic homogeneous media, which can be reduced to corresponding 2D ones. For their solution, we propose and compare two boundary integral equation approaches, both based on the coupling of a discrete time convolution quadrature with a classical space collocation discretization. In the first approach, the PDE problem is preliminarily replaced by the equivalent well known (vector) space–time boundary integral equation formulation, while in the second, the same PDE is replaced by a system of two (coupled) wave equations, each one of which is then represented by the associated boundary integral equation. The construction of these two approaches is described and discussed. Some numerical testing are also presented.

We consider a wave propagation problem in 2D, reformulated in terms of a Boundary Integral Equation (BIE) in the space-time domain. For its solution, we propose a numerical scheme based on a convolution quadrature formula by Lubich for the discretization in time, and on a Galerkin method in space. It is known that the main advantage of Lubich’s formulas is the use of the FFT algorithm to retrieve discrete time integral operators with a computational complexity of order R log R , R being twice the total number of time steps performed. Since the discretization in space leads in general to a quadratic complexity, the global computational complexity is of order M 2 R log R and the working storage required is M 2 R / 2 , where M is the number of grid points on the domain boundary. To reduce the complexity in space, we consider here approximant functions of wavelet type. By virtue of the properties of wavelet bases, the discrete integral operators have a rapid decay to zero with respect to time, and the overwhelming majority of the associated matrix entries assume negligible values. Based on an a priori estimate of the decaying behaviour in time of the matrix entries, we devise a time downsampling strategy that allows to compute only those elements which are significant with respect to a prescribed tolerance. Such an approach allows to retrieve the temporal history of each entry e (corresponding to a fixed couple of wavelet basis functions), via a Fast Fourier Transform, with computational complexity of order R ¯ e log R ¯ e . The parameter R ¯ e depends on the two basis functions, and it satisfies R ¯ e ≪ R for a relevant percentage of matrix entries, percentage which increases significantly as time and/or space discretization are refined. Globally, the numerical tests show that the computational complexity and memory storage of the overall procedure are linear in space and time for small velocities of the wave propagation, and even sub-linear for high velocities.

In this paper, we consider 3D wave propagation problems in unbounded domains, such as those of acoustic waves in non viscous fluids, or of seismic waves in (infinite) homogeneous isotropic materials, where the propagation velocity c is much higher than 1. For example, in the case of air and water c ≈ 343 m / s and c ≈ 1500 m / s respectively, while for seismic P-waves in linear solids we may have c ≈ 6000 m / s or higher. These waves can be generated by sources, possible away from the obstacles. We further assume that the dimensions of the obstacles are much smaller than that of the wave velocity, and that the problem transients are not excessively short. For their solution we consider two different approaches. The first directly uses a known space–time boundary integral equation to determine the problem solution. In the second one, after having defined an artificial boundary delimiting the region of computational interest, the above mentioned integral equation is interpreted as a non reflecting boundary condition to be coupled with a classical finite element method. For such problems, we show that in some cases the computational cost and storage, required by the above numerical approaches, can be significantly reduced by taking into account a property that till now has not been considered. To show the effectiveness of this reduction, the proposed approach is applied to several problems, including multiple scattering.

In this paper we consider the 2D exterior problem for the non homogeneous wave equation, with mixed type boundary conditions and non homogeneous initial conditions, these latter having local supports. First we derive a space-time boundary integral equation formulation to solve the problem. Then we propose a numerical approach for its solution, which couples a Lubich discrete convolution quadrature with a classical collocation method. For the computation of the extra "volume" integrals generated by the initial data, having local supports, we propose a new numerical approach. Finally, we solve some test problems and present some of the numerical results we have obtained.

In this paper we examine the computation of the potential generated by space-time BIE representations associated with Dirichlet and Neumann problems for the 2D wave equation. In particular, we consider the efficient evaluation of the (convolution) time integral that appears in the potential representation. For this, we propose two simple quadrature rules which appear more efficient than the currently used ones. Both are of Gaussian type: the classical Gauss-Jacobi quadrature rule in the case of a Dirichlet problem, and the classical Gauss-Radau quadrature rule in the case of a Neumann problem. Both of them give very accurate results by using a few quadrature nodes, as long as the potential is evaluated at a point not very close to the boundary of the PDE problem domain. To deal with this latter case, we propose an alternative rule, which is defined by a proper combination of a Gauss-Legendre formula with one of product type, the latter having only 5 nodes. The proposed quadrature formulas are compared with the second order BDF Lubich convolution quadrature formula and with two higher order Lubich formulas of Runge-Kutta type. An extensive numerical testing is presented. This shows that the new proposed approach is very competitive, from both the accuracy and efficiency points of view.

In the last years several authors have used Lubich convolution quadrature formulas to discretize space-time boundary integral equations representing time dependent problems. These rules have the fundamental property of not using explicitly the expression of the kernel of the integral equation they are applied to, which is instead replaced by that of its Laplace transform, usually given by a simple analytic function. In this paper, a review of these rules, which includes their main properties, several new remarks and some conjectures, will be presented when they are applied to the heat and wave space-time boundary integral equation formulations. The construction and behavior of the corresponding coefficients are analyzed and tested numerically. When the quadrature is defined by a BDF method, a new approach for the representation of its coefficients is presented.

In this paper we propose an efficient strategy to compute nearly singular integrals over planar triangles in R^3, arising in 3D Galerkin BEM. The strategy is based on a proper use of various nonlinear transformations, which smooth, or move away or quite eliminate all the singularities close to the domain of integration. We will deal with near singularities of the form 1/r and 1/r^3, [email protected][email protected]? being the distance between two generic points x and y lying on triangles which are disjoint, but very close to each other. The approach proposed here is demonstrated by numerical experiments.

In this paper, we propose an efficient strategy to compute nearly singular integrals over planar triangles in R3 arising in boundary element method collocation. The strategy is based on a proper use of various non-linear transformations, which smooth or move away or quite eliminate all the singularities close to the domain of integration. We will deal with near singularities of the form 1/r, 1/r2 and 1/r3, r=∥x−y∥ being the distance between a fixed near observation point x and a generic point y of a triangular element. Extensive numerical tests and comparisons with some already existing methods show that the approach proposed here is highly efficient and competitive. Copyright © 2007 John Wiley & Sons, Ltd.

We consider the retarded potential boundary integral equation, arising from the 3D Dirichlet exterior wave equation problem. For its numerical solution we use compactly supported temporal basis functions in time and a standard collocation method in space. Since the accurate computation of the integrals involved in the numerical scheme is a key issue for the numerical stability, we propose a new efficient and competitive quadrature strategy. We compare this approach with the one that uses the Lubich time convolution quadrature, and show pros and cons of both methods.

We are concerned with the numerical solution of the generalized airfoil equation for an airfoil with a flap by means of global algebraic polynomial approximants. This problem has been considered in a recent paper by Monegato and Sloan [SIAM J. Numer. Anal., 34 (1997), pp. 2288--2305], where the authors proved the stability and convergence of a Galerkin method based on high order polynomials. They also presented very promising numerical results for the corresponding collocation method but left the theoretical results of stability and convergence as an open problem. We have succeeded in proving the stability and convergence properties of this collocation method by slightly modifying it in the neighborhood of the flap. The rate of convergence we have derived for our collocation method is very similar to that of the above Galerkin method.

Several quadrature rules for the numerical integration of smooth (nonoscillatory) functions, defined on the real (positive) semiaxis or on the real axis and decaying algebraically at infinity, are examined. Among those considered for the real axis, four alternative numerical approaches are new. The advantages and the drawbacks of each of them are pointed out through several numerical tests, either on the computation of a single integral or on the numerical solution of some integral equations.

In this paper we propose special strategies to compute 1D integrals of functions having weakly or strong singularities at the endpoints of the interval of integration or complex poles close to the domain of integration. As application of the proposed strategies, we compute a four dimensional integral arising from 3D Galerkin boundary element methods (BEM) applied to hypersingular boundary integral equations.

The aim of this paper is to show the efficiency of the use of smoothing changes of variable in the numerical treatment of 1D and 2D weakly singular and singular integral equations. The introduction of a smoothing transformation, besides smoothing the solution, allows also the use of a very simple and efficient collocation method based on Chebyshev polynomials of the first kind and their zeros. Further, we propose proper smoothing changes of variable also for the numerical approximation of those collocation matrix elements, which are given by weakly singular, singular or nearly singular integrals. Several numerical tests are given to point out the efficiency of the numerical approach we propose. Copyright © 2003 John Wiley & Sons, Ltd.

In this paper we deal with the problem of constructing efficient rules for the numerical evaluation of integrals of functions which are very smooth everywhere in the domain of integration, except at the boundaries where they possess mild singularities. In particular, we consider integrals defined on bounded intervals or on triangles. Integrals of this type appear, for example, in the numerical solution of singular and weakly singular integral equations by boundary element methods.

To solve one-dimensional linear weakly singular integral equations on bounded intervals, with input functions which may be smooth or not, we propose to introduce first a simple smoothing change of variable, and then to apply classical numerical methods such as product-integration and collocation based on global polynomial approximations. The advantage of this approach is that the order of the methods can be arbitrarily high and that the associated linear systems one has to solve are very well-conditioned.

The authors develop an algorithm for the numerical evaluation of the finite Hilbert transform, with respect to non-standard weight functions, by a product quadrature rule. In particular, this algorithm allows us to deal with the weight functions with algebraic and/or logarithmic singularities in the interval [-1, 1], by using the Chebyshev points as quadrature nodes. The practical application of the rule is shown to be straightforward and to yield satisfactory numerical results. Convergence theorems are also given, when the nodes are the zeros of certain classical Jacobi polynomials and the weight is defined as a generalized Ditzian-Totik weight.