Equivalent Partial Differential Equation Research Articles

Modelling of the advection–diffusion equation with a meshless method without numerical diffusion

A comprehensive study is presented regarding the stability of the forward explicit integration technique with generalized finite-difference spatial discretizations, free of numerical diffusion, applied to the advection–diffusion equation. The modified equivalent partial differential equation approach is used to demonstrate that the approximation is free of numerical diffusion. Two-dimensional results are obtained using the von Neumann method of stability analysis. Numerical results are presented showing the accuracy obtained.

Analysis and higher-order extension of the VC2 confinement scheme

This paper presents a detailed analysis of the VC2 confinement scheme for the linear transport equation. The confinement appears as a nonlinear second-order difference which drives the solution towards stable pulses when it operates in a globally anti-diffusive manner. Using the equivalent partial differential equation approach, both the confinement and the numerical dissipation of the discretization introduce forcing terms into the original equation, which govern the behavior of the solution for very large times. In a frame moving at the convection speed of the underlying equation, the signal is found to relax towards an asymptotic solution satisfying a fixed-point nonlinear difference equation expressing the balance between compression and diffusion. Based on the same ideas, a generalization of the VC2 confinement scheme to higher-orders is derived. It works for any odd-order scheme and is obtained by using a nonlinear mean between a centered and an upwind estimate of the lower-order dissipative term of the scheme. The asymptotic solution is the same for all orders of discretization and depends on the confinement scheme only. The solution of the transport equation relaxes towards this asymptotic solution at a rate depending on the order of the scheme. Numerical tests for various sets of initial conditions support the above findings.

Nonlinear ordinary differential equations in fluid dynamics

The equivalence between nonlinear ordinary differential equations (ODEs) and linear partial differential equations (PDEs) was recently revisited by Smith, who used the equivalence to transform the ODEs of Newtonian dynamics into equivalent PDEs, from which analytical solutions to several simple dynamical problems were derived. We show how this equivalence can be used to derive a variety of exact solutions to the PDEs describing advection in fluid dynamics in terms of solutions to the equivalent ODEs for the trajectories of Lagrangian fluid particles. The PDEs that we consider describe the time evolution of non-diffusive scalars, conserved densities, and Lagrangian surfaces advected by an arbitrary compressible fluid velocity field u(x, t). By virtue of their arbitrary initial conditions, the analytical solutions are asymmetric and three-dimensional even when the velocity field is one-dimensional or symmetrical. Such solutions are useful for verifying multidimensional numerical algorithms and computer codes for simulating advection and interfacial dynamics in fluids. Illustrative examples are discussed.

The differential equation counterpart of an individual-based model for yeast population growth

Computer simulations are increasingly used in biological fields. The amazing power, storage ability, and processing speeds available nowadays have enabled the implementation of computer individual-based models (IbMs) to simulate really complex biological populations. Computers can easily keep track of thousands of individuals (often called ’agents’), whose complex behaviours and large amounts of associated data were daunting only 20 years ago. As such, computer modelling has just entered a field where traditional PDE models used to reign alone. A study of the exchange and non-trivial relationship between these two fields, computer IbMs versus classic partial differential equations (PDEs), is appropriate. The aim of this paper is to compare both approaches through a relevant example, namely the evolution of a yeast population in a batch culture. Thus, this paper deals with the utilization of both classical mathematics and computer science in the solution of problems arising in microbiology. First, an IbM approach to study the evolution of a yeast batch culture is presented. Second, an equivalent PDE model is derived by using some techniques from the interacting particle systems field. Third, a comparison and discussion on the advantages and drawbacks of both modelling tools is given.

Towards higher order lattice Boltzmann schemes

In this contribution we extend the Taylor expansion method proposed previously by one ofus and establish equivalent partial differential equations for the lattice Boltzmann schemeproposed by d’Humières (1992 Rarefied Gas Dynamics: Theory and Simulations (AIAAProgress in Astronautics and Astronautics vol 159) pp 450–8) at an arbitrary order ofaccuracy. We derive formally the associated dynamical equations for classical thermal andlinear fluid models in one to three space dimensions. We use this approach to adjust‘quartic’ relaxation parameters in order to enforce fourth-order accuracy for the thermalmodel and diffusive relaxation modes of the Stokes problem. We apply the resulting schemefor the numerical computation of the associated eigenmodes, compare our resultswith analytical references and observe fourth-order accuracy when using ‘quartic’parameters.

Third order equivalent equation of lattice Boltzmann scheme

We recall the origin of lattice Boltzmann scheme and detail the version due to D'Humières [8]. We present a formal analysisof this lattice Boltzmann scheme in terms of a single numericalinfinitesimal parameter. We derive third order equivalent partial differential equation of this scheme.Both situations of single conservation law and fluid flow with mass and momentum conservations are detailed. We apply our analysis to so-called D1Q3 and D2Q9 lattice Boltzmann schemes in one and two space dimensions.

Modeling electrocortical activity through improved local approximations of integral neural field equations

Neural field models of firing rate activity typically take the form of integral equations with space-dependent axonal delays. Under natural assumptions on the synaptic connectivity we show how one can derive an equivalent partial differential equation (PDE) model that properly treats the axonal delay terms of the integral formulation. Our analysis avoids the so-called long-wavelength approximation that has previously been used to formulate PDE models for neural activity in two spatial dimensions. Direct numerical simulations of this PDE model show instabilities of the homogeneous steady state that are in full agreement with a Turing instability analysis of the original integral model. We discuss the benefits of such a local model and its usefulness in modeling electrocortical activity. In particular, we are able to treat "patchy" connections, whereby a homogeneous and isotropic system is modulated in a spatially periodic fashion. In this case the emergence of a "lattice-directed" traveling wave predicted by a linear instability analysis is confirmed by the numerical simulation of an appropriate set of coupled PDEs.

Equivalent partial differential equations of a lattice Boltzmann scheme

We show that when we formulate the lattice Boltzmann equation with a small-time step Δ t and an associated space scale Δ x , a Taylor expansion joined with the so-called equivalent equation methodology leads to establishing macroscopic fluid equations as a formal limit. We recover the Euler equations of gas dynamics at the first order and the compressible Navier–Stokes equations at the second order.

Une introduction au schéma de Boltzmann sur réseau

We propose an elementary introduction to the lattice Boltzmann scheme. We recall the physical (Boltzmann equation) and algorithmic (cellular automata) origins of this numerical method. For a one-dimensional example, we present in detail the two characteristic steps of the algorithm: the nonlinear collision step, local in space and the linear propagation phase with the neighbouring vertices, explicit in time. We then propose a generic Taylor-type development with the so-called equivalent partial differential equation. We obtain in this way formally a so-called Chapman-Enskog development where the small parameter is the discretization step of the scheme. At order zero, the lattice Boltzmann scheme satisfies a local thermodynamical equilibrium. At first order, it satisfies the Euler equations of gas dynamics and at second order the Navier-Stokes equations. Then we detail the classical case of the nine velocities model on a square lattice.

Numerical solutions of diffusive logistic equation

Abstract In this paper we investigate numerically positive solutions of a superlinear Elliptic equation on bounded domains. The study of Diffusive logistic equation continues to be an active field of research. The subject has important applications to population migration as well as many other branches of science and engineering. In this paper the “ finite difference scheme ” will be developed and compared for solving the one- and three-dimensional Diffusive logistic equation. The basis of the analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from many authors these years.

The axisymmetric consolidation of a semi‐infinite transversely isotropic saturated soil

AbstractThis paper presents an exact analytical solution to fully coupled axisymmetric consolidation of a semi‐infinite, transversely isotropic saturated soil subjected to a uniform circular loading at the ground surface. The analysis is under the framework of Biot's general theory of consolidation. First, the governing equations of consolidation are transformed into a set of equivalent partial differential equations with the introduction of two auxiliary variables. These partial differential equations are then solved using Hankel–Laplace integral transforms. Once solutions in the transformed domain have been obtained, the actual solutions in the physical domain for displacements and stress components of the solid matrix, pore‐water pressure and fluid discharge can be finally obtained by direct numerical inversion. The accuracy of the numerical solutions developed is confirmed by comparison with an existing exact solution for an isotropic and saturated soil that is a special case of the more general problem addressed. Numerical analyses are also presented to investigate the influence of the degree of material anisotropy on the consolidation settlement. Copyright © 2005 John Wiley & Sons, Ltd.

On the numerical solution of the one‐dimensional convection‐diffusion equation

The numerical solution of convection‐diffusion transport problems arises in many important applications in science and engineering. These problems occur in many applications such as in the transport of air and ground water pollutants, oil reservoir flow, in the modeling of semiconductors, and so forth. This paper describes several finite difference schemes for solving the one‐dimensional convection‐diffusion equation with constant coefficients. In this research the use of modified equivalent partial differential equation (MEPDE) as a means of estimating the order of accuracy of a given finite difference technique is emphasized. This approach can unify the deduction of arbitrary techniques for the numerical solution of convection‐diffusion equation. It is also used to develop new methods of high accuracy. This approach allows simple comparison of the errors associated with the partial differential equation. Various difference approximations are derived for the one‐dimensional constant coefficient convection‐diffusion equation. The results of a numerical experiment are provided, to verify the efficiency of the designed new algorithms. The paper ends with a concluding remark.

Quasi-implicit and two-level explicit finite-difference procedures for solving the one-dimensional advection equation

A variety of explicit and implicit algorithms has been studied dealing with the solution of the one-dimensional advection equation. These schemes are based on the weighted finite difference approximations. This solution approach proves to be very effective. This procedure is carried out in many disciplines including hydrology, oceanography and meteorology. The main idea behind the finite-difference methods for obtaining the solution of a given partial differential equation is to approximate the derivatives appearing in the equation by a set of values of the function at a selected number of points. The most usual way to generate these approximations is through the use of Taylor series. The basis of analysis of the finite-difference equations considered here is the modified equivalent partial differential equation approach. It is worth noting that from the truncation error of the modified equivalent equation, it is possible to eliminate the dominant error terms associated with the finite-difference equations that contain weights, thus leading to more accurate techniques. Quasi-implicit and two-level explicit finite-difference techniques are considered in this article to approximately solve the one-dimensional advection problem. Stability of the techniques are investigated using the von Neumann stability analysis. The results of a numerical experiment are presented, and the accuracy and central processor unit (CPU) time needed are discussed and compared. Numerical tests show that the new third-order or fourth-order methods are more accurate than either of the second-order techniques.

On the solution of an initial‐boundary value problem that combines Neumann and integral condition for the wave equation

AbstractNumerical solution of hyperbolic partial differential equation with an integral condition continues to be a major research area with widespread applications in modern physics and technology. Many physical phenomena are modeled by nonclassical hyperbolic boundary value problems with nonlocal boundary conditions. In place of the classical specification of boundary data, we impose a nonlocal boundary condition. Partial differential equations with nonlocal boundary specifications have received much attention in last 20 years. However, most of the articles were directed to the second‐order parabolic equation, particularly to heat conduction equation. We will deal here with new type of nonlocal boundary value problem that is the solution of hyperbolic partial differential equations with nonlocal boundary specifications. These nonlocal conditions arise mainly when the data on the boundary can not be measured directly. Several finite difference methods have been proposed for the numerical solution of this one‐dimensional nonclassic boundary value problem. These computational techniques are compared using the largest error terms in the resulting modified equivalent partial differential equation. Numerical results supporting theoretical expectations are given. Restrictions on using higher order computational techniques for the studied problem are discussed. Suitable references on various physical applications and the theoretical aspects of solutions are introduced at the end of this article. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005

Robust Biased Brownian Dynamics for Rate Constant Calculation

A reaction probability is required to calculate the rate constant of a diffusion-dominated reaction. Due to the complicated geometry and potentially high dimension of the reaction probability problem, it is usually solved by a Brownian dynamics simulation, also known as a random walk or path integral method, instead of solving the equivalent partial differential equation by a discretization method. Building on earlier work, this article completes the development of a robust importance sampling algorithm for Brownian dynamics—i.e., biased Brownian dynamics with weight control—to overcome the high energy and entropy barriers in biomolecular association reactions. The biased Brownian dynamics steers sampling by a bias force, and the weight control algorithm controls sampling by a target weight. This algorithm is optimal if the bias force and the target weight are constructed from the solution of the reaction probability problem. In reality, an approximate reaction probability has to be used to construct the bias force and the target weight. Thus, the performance of the algorithm depends on the quality of the approximation. Given here is a method to calculate a good approximation, which is based on the selection of a reaction coordinate and the variational formulation of the reaction probability problem. The numerically approximated reaction probability is shown by computer experiments to give a factor-of-two speedup over the use of a purely heuristic approximation. Also, the fully developed method is compared to unbiased Brownian dynamics. The tests for human superoxide dismutase, Escherichia coli superoxide dismutase, and antisweetener antibody NC6.8, show speedups of 17, 35, and 39, respectively. The test for reactions between two model proteins with orientations shows speedups of 2578 for one set of configurations and 3341 for another set of configurations.

Numerical solution of the three-dimensional advection–diffusion equation

The study of advection–diffusion equation continues to be an active field of research. The subject has important applications to fluid dynamics as well as many other branches of science and engineering. In this paper several different numerical techniques will be developed and compared for solving the three-dimensional advection–diffusion equation with constant coefficient. These techniques are based on the two-level fully explicit and fully implicit finite difference approximations. The basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyett. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes. Another nice feature of the modified equivalent partial differential equation approach is that a high order of accuracy can be combined with excellent stability properties. The new second-order accurate methods are free of numerical diffusion. The results of a numerical experiment are presented, and the accuracy and central processor (CPU) time needed are discussed and compared.

Weighted finite difference techniques for the one-dimensional advection–diffusion equation

Various numerical techniques will be developed and compared for solving the one-dimensional advection–diffusion equation with constant coefficient. These techniques are based on the two-level finite difference approximations. The basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyett. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes. The new methods are more accurate and are more efficient than the conventional techniques. These schemes are free of numerical diffusion. The results of a numerical experiment are presented, and the accuracy and central processor (CPU) time needed are discussed and compared.

Waves and bumps in neuronal networks with axo-dendritic synaptic interactions

We consider a firing rate model of a neuronal network continuum that incorporates axo-dendritic synaptic processing and the finite conduction velocities of action potentials. The model equation is an integral one defined on a spatially extended domain. Apart from a spatial integral mixing the network connectivity function with space-dependent delays, arising from non-instantaneous axonal communication, the integral model also includes a temporal integration over some appropriately identified distributed delay kernel. These distributed delay kernels are biologically motivated and represent the response of biological synapses to spiking inputs. They are interpreted as Green’s functions of some linear differential operator. Exploiting this Green’s function description we discuss formal reductions of this non-local system to equivalent partial differential equation (PDE) models. We distinguish between those spatial connectivity functions that give rise to local PDE models and those that give rise to PDE models with delayed non-local terms. For cases in which local PDEs are derived, we investigate traveling wave solutions in a comoving frame by numerically computing global heteroclinic connections for sigmoidal firing rate functions. We also calculate exact solutions, parameterized by axonal conduction velocity, for the Heaviside firing rate function (the sigmoidal firing rate function in the limit of infinite gain). The inclusion of synaptic adaptation is shown to alter traveling wave fronts to traveling pulses, which we study analytically and numerically in terms of a global homoclinic orbit. Finally, we consider the impact of dendritic interactions on waves and on static spatially localized solutions. Exact analysis for infinite gain shows that axonal delays do not affect the stability of single bumps. Furthermore, numerical continuation for finite gain leads to multiple bump solutions, and it is demonstrated that such localized multi-bumps are lost (in favor of global patterns) when a stable N-bump and an unstable ( N+2)-bump coalesce. In essence, the work in this paper illustrates how physiologically significant features of synaptic processing, synaptic adaptation and patterns of axo-dendritic connectivity may be analyzed within a neural field firing rate framework.

Numerical solution of a non‐local boundary value problem with Neumann's boundary conditions

AbstractSeveral second‐order finite difference schemes are discussed for solving a non‐local boundary value problem for two‐dimensional diffusion equation with Neumann's boundary conditions. While sharing some common features with the one‐dimensional models, the solution of two‐dimensional equations are substantially more difficult, thus some considerations are taken to be able to extend some ideas of one‐dimensional case. Using a suitable transformation the solution of this problem is equivalent to the solution of two other problems. The former which is a one‐dimensional non‐local boundary value problem gives the value of μthrough using the unconditionally stable standard implicit (3,1) backward time centred space (denoted BTCS) scheme. Using this result the second problem will be changed to a classical two‐dimensional diffusion equation with Neumann's boundary conditions which will be solved numerically by using two unconditionally stable fully implicit finite difference schemes, or using two conditionally stable fully explicit finite difference techniques. For each method investigated the modified equivalent partial differential equation is employed which permits the order of accuracy of the numerical techniques to be determined. The results of a numerical example for all finite difference schemes discussed in this paper are given and computation times are presented. Copyright © 2003 John Wiley & Sons, Ltd.

A new ADI technique for two-dimensional parabolic equation with an integral condition

Nonclassical parabolic initial-boundary value problems arise in the study of several important physical phenomena. This paper presents a new approach to treat complicated boundary conditions appearing in the parabolic partial differential equations with nonclassical boundary conditions. A new fourth-order finite difference technique, based upon the Noye and Hayman (N-H) alternating direction implicit (ADI) scheme, is used as the basis to solve the two-dimensional time dependent diffusion equation with an integral condition replacing one boundary condition. This scheme uses less central processor time (CPU) than a second-order fully implicit scheme based on the classical backward time centered space (BTCS) method for two-dimensional diffusion. It also has a larger range of stability than a second-order fully explicit scheme based on the classical forward time centered space (FTCS) method. The basis of the analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyeet. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference methods. The results of numerical experiments for the new method are presented. The central processor times needed are also reported. Error estimates derived in the maximum norm are tabulated.