Time-dependent Partial Differential Equations Research Articles

Tensor numerical methods for multidimensional PDES: theoretical analysis and initial applications

We present a brief survey on the modern tensor numerical methods for multidimen- sional stationary and time-dependent partial differential equations (PDEs). The guiding principle of the tensor approach is the rank-structured separable approximation of multivariate functions and operators represented on a grid. Recently, the traditional Tucker, canonical, and matrix product states (tensor train) tensor models have been applied to the grid-based electronic structure cal- culations, to parametric PDEs, and to dynamical equations arising in scientific computing. The essential progress is based on the quantics tensor approximation method proved to be capable to represent (approximate) function related d-dimensional data arrays of size N d with log-volume com- plexity, O(dlogN). Combined with the traditional numerical schemes, these novel tools establish a new promising approach for solving multidimensional integral and differential equations using low-parametric rank-structured tensor formats. As the main example, we describe the grid-based tensor numerical approach for solving the 3D nonlinear Hartree-Fock eigenvalue problem, that was the starting point for the developments of tensor-structured numerical methods for large-scale com- putations in solving real-life multidimensional problems. We also discuss a new method for the fast 3D lattice summation of electrostatic potentials by assembled low-rank tensor approximation capable to treat the potential sum over millions of atoms in few seconds. We address new results on tensor approximation of the dynamical Fokker-Planck and master equations in many dimensions up to d = 20. Numerical tests demonstrate the benefits of the rank-structured tensor approximation on the aforementioned examples of multidimensional PDEs. In particular, the use of grid-based tensor representations in the reduced basis of atomics orbitals yields an accurate solution of the Hartree-Fock equation on large N × N × N grids with a grid size of up to N = 10 5 .

Comparison of Finite Difference Schemes for the Wave Equation Based on Dispersion

Finite difference techniques are widely used for the numerical simulation of time-dependent partial differential equations. In order to get better accuracy at low computational cost, researchers have attempted to develop higher order methods by improving other lower order methods. However, these types of methods usually suffer from a high degree of numerical dispersion. In this paper, we review three higher order finite difference methods, higher order compact (HOC), compact Padé based (CPD) and non-compact Padé based (NCPD) schemes for the acoustic wave equation. We present the stability analysis of the three schemes and derive dispersion characteristics for these schemes. The effects of Courant Friedrichs Lewy (CFL) number, propagation angle and number of cells per wavelength on dispersion are studied.

Unsteady Radiative-Convective Boundary-Layer Flow of a Casson Fluid with Variable Thermal Conductivity

The unsteady two-dimensional flow of a non-Newtonian fluid over a stretching surface with the effects of thermal radiation and variable thermal conductivity is investigated. The Casson fluid model is used to characterize the non-Newtonian fluid behavior. First, using a similarity transformation, the governing time-dependent partial differential equations are transformed into coupled nonlinear ordinary differential equations with variable coefficients. Then the transformed equations are solved numerically under appropriate boundary conditions by the shooting method. An exact solution corresponding to the momentum equation for a steady case is found. The obtained numerical results are analyzed as to the effect of the pertinent parameters on the flow and heat transfer characteristics.

A local discontinuous Galerkin method based on variational structure

We present a special variant of the local discontinuous Galerkin (LDG) method for time-dependent partial differential equations with certain variational structures and associated conservation or dissipation properties. The method provides a way to construct fully-discrete LDG schemes that retain discrete counterparts of the conservation or dissipation properties. Numerical results confirm the accuracy and effectiveness of the method.

Towards efficient backward-in-time adjoint computations using data compression techniques

Abstract In the context of a posteriori error estimation for nonlinear time-dependent partial differential equations, the state-of-the-practice is to use adjoint approaches which require the solution of a backward-in-time problem defined by a linearization of the forward problem. One of the major obstacles in the practical application of these approaches is the need to store, or recompute, the forward solution to define the adjoint problem and to evaluate the error representation. This study considers the use of data compression techniques to approximate forward solutions employed in the backward-in-time integration. The development derives an error representation that accounts for the difference between the standard-approach and the compressed approximation of the forward solution. This representation is algorithmically similar to the standard representation and only requires the computation of the quantity of interest for the forward solution and the data-compressed reconstructed solution (i.e. scalar quantities that can be evaluated as the forward problem is integrated). This approach is then compared with existing techniques, such as checkpointing and time-averaged adjoints. Finally, we provide numerical results indicating the potential efficiency of our approach on a transient diffusion–reaction equation and on the Navier–Stokes equations. These results demonstrate memory compression ratios up to 450 × while maintaining reasonable accuracy in the error-estimates.

Numerical simulations of time-dependent partial differential equations

When a time-dependent partial differential equation (PDE) is discretized in space with a spectral approximation, the result is a coupled system of ordinary differential equations (ODEs) in time. This is the notion of the method of lines (MOL), and the resulting set of ODEs is stiff; the stiffness may be even exacerbated sometimes. The linear terms are the primarily responsible for the stiffness, with a rapid exponential decay of some modes (as in a dissipative PDE), or a rapid oscillation of some modes (as in a dispersive PDE). Therefore, for a time-dependent PDE which combines low-order nonlinear terms with higher-order linear terms, it is desirable to use a higher-order approximation both in space and in time.Along our research, we have focused on a particular case of spectral methods, the so-called pseudo-spectral methods, to solve numerically time-dependent PDEs using different techniques: an integrating factor, in de la Hoz and Vadillo (2010); an exponential time differencing method, in de la Hoz and Vadillo (2008); and differentiation matrices in the theoretical frame of matrix differential equations, in de la Hoz and Vadillo (2012, 2013a,b). This paper, which is a unified review of those contributions, aims at providing a better understanding of those methods, by illustrating their variety and, more importantly, their power. Furthermore, we also give emphasis to choosing adequate schemes to advance in time.

A simple moving mesh method for blowup problems

We develop a simple and efficient adaptive mesh generator for time-dependent partial differential equations with singular solutions in two dimensional spaces. The mesh generator is based on minimizing the sum of two diagonal lengths in each cell. We also add second order difference terms to obtain smoother and more orthogonal mesh. The method is successfully applied to the nonlinear heat equations with blowup solutions. We can obtain a solution with an amplitude of 1015 at the peak and the mesh difference of 10?16 near the peak. We also discuss nonlinear heat equations whose solutions blow up at space infinity and whose blowup time is given.

Numerical solution of time-dependent diffusion equations with nonlocal boundary conditions via a fast matrix approach

Abstract This article contributes a matrix approach by using Taylor approximation to obtain the numerical solution of one-dimensional time-dependent parabolic partial differential equations (PDEs) subject to nonlocal boundary integral conditions. We first impose the initial and boundary conditions to the main problems and then reach to the associated integro-PDEs. By using operational matrices and also the completeness of the monomials basis, the obtained integro-PDEs will be reduced to the generalized Sylvester equations. For solving these algebraic systems, we apply a famous technique in Krylov subspace iterative methods. A numerical example is considered to show the efficiency of the proposed idea.

Numerical analysis of energy-minimizing wavelengths of equilibrium states for diblock copolymers

We present a robust and accurate numerical algorithm for calculating energy-minimizing wavelengths of equilibrium states for diblock copolymers. The phase-field model for diblock copolymers is based on the nonlocal Cahn–Hilliard equation. The model consists of local and nonlocal terms associated with short- and long-range interactions, respectively. To solve the phase-field model efficiently and accurately, we use a linearly stabilized splitting-type scheme with a semi-implicit Fourier spectral method. To find energy-minimizing wavelengths of equilibrium states, we take two approaches. One is to obtain an equilibrium state from a long time simulation of the time-dependent partial differential equation with varying periodicity and choosing the energy-minimizing wavelength. The other is to directly solve the ordinary differential equation for the steady state. The results from these two methods are identical, which confirms the accuracy of the proposed algorithm. We also propose a simple and powerful formula: h = L∗/m, where h is the space grid size, L∗ is the energy-minimizing wavelength, and m is the number of the numerical grid steps in one period of a wave. Two- and three-dimensional numerical results are presented validating the usefulness of the formula without trial and error or ad hoc processes.

Runge–Kutta Residual Distribution Schemes

We are concerned with the solution of time-dependent non-linear hyperbolic partial differential equations. We investigate the combination of residual distribution methods with a consistent mass matrix (discretisation in space) and a Runge–Kutta-type time-stepping (discretisation in time). The introduced non-linear blending procedure allows us to retain the explicit character of the time-stepping procedure. The resulting methods are second order accurate provided that both spatial and temporal approximations are. The proposed approach results in a global linear system that has to be solved at each time-step. An efficient way of solving this system is also proposed. To test and validate this new framework, we perform extensive numerical experiments on a wide variety of classical problems. An extensive numerical comparison of our approach with other multi-stage residual distribution schemes is also given.

Sextic B-spline collocation method for solving Euler–Bernoulli Beam Models

A numerical method based on sextic B-spline is developed to solve the fourth-order time-dependent partial differential equations subjected to fixed and cantilever boundary conditions. We use finite difference approximation to discretize the temporal variable and the spatial variable by means of a σ-method, σ∈[0,1] (σ=12 corresponds to the Crank–Nicolson method), and a sextic B-spline collocation method on uniform meshes, respectively. Using Von Neumann method, the proposed method is also shown to be conditionally stable if σ<0.25 and unconditionally stable if σ⩾0.25. The convergence analysis of the proposed sextic B-spline approximation for the Euler–Bernoulli problem is discussed in details and we have shown under appropriate conditions the proposed method converges. Some physical examples and their numerical results are provided to justify the advantages of the proposed method.

Finite block method for transient heat conduction analysis in functionally graded media

SUMMARYBased on the one‐dimensional differential matrix derived from the Lagrange series interpolation, the finite block method is proposed first time to solve both stationary and transient heat conduction problems of anisotropic and functionally graded materials. The main idea is to establish the first order one‐dimensional differential matrix constructed by using Lagrange series with uniformly distributed nodes. Then the higher order of derivative matrix for one‐dimensional problem is obtained. By introducing the mapping technique, a block of quadratic type is transformed from Cartesian coordinate (xyz) to normalised coordinate (ξης) with 8 seeds or 20 seeds for two or three dimensions. Then the differential matrices in physical domain are determined from that in normalised transformed coordinate system. In addition, the time dependent partial differential equations are analysed in the Laplace transformed domain, and the Durbin inversion method is used to determine the values in time domain. Illustrative two‐dimensional and three‐dimensional numerical examples are given, and comparisons have been made with analytical solutions. Copyright © 2014 John Wiley &amp; Sons, Ltd.

A residual based error estimate for Leja interpolation of matrix functions

Recent applications in science and engineering require a reliable implementation of the action of the matrix exponential and related φ functions. For possibly large matrices an approximation up to a certain tolerance needs to be obtained in a robust and efficient way. In this work we consider Leja interpolation for performing this task. Our focus lies on a new a posteriori error estimate for the method. We introduce the notion of a residual based estimate, where the residual is obtained from differential equations defining the φ functions. The properties of this new error estimate are investigated and compared to an existing one. Further, a numerical investigation is performed based on test examples originating from spatial discretization of time dependent partial differential equations in two and three space dimensions. The experiments show that this new approach is robust for various types of matrices and applications.

Computer Modelling of RFA Hepatic Tumor Ablation in 3D

The minimally invasive treatment called radio-frequency ablation(RFA) guided by imaging techniques, the doctor inserts a thin needlethrough the skin and into the tumor. High-frequency electrical energydelivered through this needle heats and destroys the tumor. Thecircuit is closed with a ground pad applied to the patient's skin. This work concerns the mathematical modeling and computer simulations of the heat transfer process. The core is solving the bio-heat time-dependent partial differential equation of parabolic type. Both, a uniform discretization of the considered time interval and an adaptive time-stepping procedure are applied. The last one is due to an effort to decrease the simulation time. Computer simulation on geometry obtained from a magnetic resonance imaging (MRI) scan of the patient is performed. In this paper we will focus on: (i) the heat transfer accounting the blood circulation in the small blood vessels, and (ii) the heat transfer accounting the blood circulation in the portal vein. Results of some numerical experiments performed on a selected test problems are presented and discussed.the model.

A general framework for finding energy dissipative/conservative $$H^1$$ H 1 -Galerkin schemes and their underlying $$H^1$$ H 1 -weak forms for nonlinear evolution equations

A general framework for constructing energy dissipative or conservative Galerkin schemes for time-dependent partial differential equations (PDEs) is presented. The framework embraces a variety of dissipative and conservative PDEs with variational structure. An advantageous feature is that implementation of the resulting scheme requires only P1 elements. The schemes and their underlying \(H^1\)-weak forms are derived from the concept of formal weak form and an \(L^2\)-projection technique.

Two Efficient Approaches Based On Radial Basis Functions To Nonlinear Time-dependent Partial Differential Equations

In this paper, two numerical meshless approaches, based on radial basis functions (RBFs) are applied to solve nonlinear time-dependent partial differential equations. Both of these approaches are based on Kansa meshless method. In these procedures the time is descritized to small time steps, the space is also descritized in each sub-domain. Then by the Kansa collocation method, by using RBFs, the approximate solution, in each step, is obtained. In the first approach, the nonlinear terms are eliminated, by linearization. In the second approach nonlinear equations are solved directly, by Fix point method. Four examples are provided to illustrate the efficiency and the reliability of these approaches. The results are also compared with each other.

Additive schemes (splitting schemes) for some systems of evolutionary equations

On the basis of additive schemes (splitting schemes) we construct efficient numerical algorithms to solve approximately the initial-boundary value problems for systems of time-dependent partial differential equations (PDEs). In many applied problems the individual components of the vector of unknowns are coupled together and then splitting schemes are applied in order to get a simple problem for evaluating components at a new time level. Typically, the additive operator-difference schemes for systems of evolutionary equations are constructed for operators coupled in space. In this paper we investigate more general problems where coupling of derivatives in time for components of the solution vector takes place. Splitting schemes are developed using an additive representation for both the primary operator of the problem and the operator at the time derivative. Splitting schemes are based on a triangular two-component representation of the operators.

Fast isogeometric solvers for explicit dynamics

In finite element analysis, solving time-dependent partial differential equations with explicit time marching schemes requires repeatedly applying the inverse of the mass matrix. For mass matrices that can be expressed as tensor products of lower dimensional matrices, we present a direct method that has linear computational complexity, i.e., O(N), where N is the total number of degrees of freedom in the system. We refer to these matrices as separable matrices. For non-separable mass matrices, we present a preconditioned conjugate gradient method with carefully designed preconditioners as an alternative. We demonstrate that these preconditioners, which are easy to construct and cheap to apply (O(N)), can deliver significant convergence acceleration. The performances of these preconditioners are independent of the polynomial order (p independence) and mesh resolution (h independence) for maximum continuity B-splines, as verified by various numerical tests.

Bifurcation from rolls to multi-pulse planforms via reduction to a parabolic Boussinesq model

A mechanism is presented for the bifurcation from one-dimensional spatially periodic patterns (rolls) into two-dimensional planar states (planforms). The novelty is twofold: the planforms are solutions of a Boussinesq partial differential equation (PDE) on a periodic background and secondly explicit formulas for the coefficients in the Boussinesq equation are derived, based on a form of planar conservation of wave action flux. The Boussinesq equation is integrable with a vast array of solutions, and an example of a new planform bifurcating from rolls, which appears to be generic, is presented. Adding in time leads to a new time-dependent PDE, which models the nonlinear behaviour emerging from a generalization of Eckhaus instability. The class of PDEs to which the theory applies is evolution equations whose steady part is a gradient elliptic PDE. Examples are the 2+1 Ginzburg–Landau equation with real coefficients, and the 2+1 planar Swift–Hohenberg equation.

A Modulus-Squared Dirichlet Boundary Condition for Time-Dependent Complex Partial Differential Equations and Its Application to the Nonlinear Schrödinger Equation

An easy to implement modulus-squared Dirichlet (MSD) boundary condition is formulated for numerical simulations of time-dependent complex partial differential equations in multidimensional settings. The MSD boundary condition approximates a constant modulus-squared value of the solution at the boundaries and is defined as $\frac{\partial \Psi}{\partial t}|_b \approx i\,\mbox{Im}[ \frac{1}{\Psi_{b-1}} \frac{\partial \Psi}{\partial t}|_{b-1} ]\,\Psi_b,$ where $\Psi$ is the complex field and the subscripts $b$ and $b-1$ refer to a boundary point and the closest interior point to the boundary, respectively. Application of the MSD boundary condition to simulations of the nonlinear Schrodinger equation is shown, and numerical simulations are performed to demonstrate its usefulness and advantages over other simple boundary conditions.