Positivity-Preserving Local Discontinuous Galerkin Method for Pattern Formation Dynamical Model in Polymerizing Actin Flocks

In this paper, we apply local discontinuous Galerkin (LDG) methods for pattern formation dynamical model in polymerizing actin flocks. There are two main difficulties in designing effective numerical solvers. First of all, the density function is non-negative, and zero is an unstable equilibrium solution. Therefore, negative density values may yield blow-up solutions. To obtain positive numerical approximations, we apply the positivity-preserving (PP) techniques. Secondly, the model may contain stiff source. The most commonly used time integration for the PP technique is the strong-stability-preserving Runge-Kutta method. However, for problems with stiff source, such time discretizations may require strictly limited time step sizes, leading to large computational cost. Moreover, the stiff source any trigger spurious filament polarization, leading to wrong numerical approximations on coarse meshes. In this paper, we combine the PP LDG methods with the semi-implicit Runge-Kutta methods. Numerical experiments demonstrate that the proposed method can yield accurate numerical approximations with relatively large time steps.     

Step size control in the numerical solution of stochastic differential equations

We introduce a variable step size algorithm for the pathwise numerical approximation of solutions to stochastic ordinary differential equations. The algorithm is based on a new pair of embedded explicit Runge-Kutta methods of strong order 1.5(1.0), where the method of strong order 1.5 advances the numerical computation and the difference between approximations defined by the two methods is used for control of the local error. We show that convergence of our method is preserved though the discretization times are not stopping times any more, and further, we present numerical results which demonstrate the effectiveness of the variable step size implementation compared to a fixed step size implementation.     

Error estimates for approximations of a gradient dynamics for phase field elastic bending energy of vesicle membrane deformation

In this paper, we study the numerical approximations of a gradient flow associated with a phase field bending elasticity model of a vesicle membrane with prescribed volume and surface area. A spatially semi‐discrete scheme based on a mixed finite element formulation and a fully discrete in space and time scheme are analyzed. Optimal order error estimates are rigorously derived for these numerical schemes without any a priori assumption. Copyright © 2013 John Wiley & Sons, Ltd.     

Uniformly exponentially stable approximations for a class of damped systems

We consider time semi-discrete approximations of a class of exponentially stable infinite-dimensional systems modeling, for instance, damped vibrations. It has recently been proved that for time semi-discrete systems, due to high frequency spurious components, the exponential decay property may be lost as the time step tends to zero. We prove that adding a suitable numerical viscosity term in the numerical scheme, one obtains approximations that are uniformly exponentially stable. This result is then combined with previous ones on space semi-discretizations to derive similar results on fully-discrete approximation schemes. Our method is mainly based on a decoupling argument of low and high frequencies, the low frequency observability property for time semi-discrete approximations of conservative linear systems and the dissipativity of the numerical viscosity on the high frequency components. Our methods also allow to deal directly with stabilization properties of fully discrete approximation schemes without numerical viscosity, under a suitable CFL type condition on the time and space discretization parameters.     

Adaptive computation for boundary control of radiative heat transfer in glass

This paper is concerned with an optimal boundary control of the cooling down process of glass, an important step in glass manufacturing. Since the computation of the complete radiative heat transfer equations is too complex for optimization purposes, we use simplified approximations of spherical harmonics including a practically relevant frequency bands model. The optimal control problem is considered as a constrained optimization problem. A first-order optimality system is derived and decoupled with the help of a gradient method based on the solution to the adjoint equations. The arising partial differential–algebraic equations of mixed parabolic–elliptic type are numerically solved by a self-adaptive method of lines approach of Rothe type. Adaptive finite elements in space and one-step methods of Rosenbrock-type with variable step sizes in time are applied. We present numerical results for a two-dimensional glass cooling problem.     

Corrected phase-type approximations of heavy-tailed risk models using perturbation analysis

Numerical evaluation of performance measures in heavy-tailed risk models is an important and challenging problem. In this paper, we construct very accurate approximations of such performance measures that provide small absolute and relative errors. Motivated by statistical analysis, we assume that the claim sizes are a mixture of a phase-type and a heavy-tailed distribution and with the aid of perturbation analysis we derive a series expansion for the performance measure under consideration. Our proposed approximations consist of the first two terms of this series expansion, where the first term is a phase-type approximation of our measure. We refer to our approximations collectively as corrected phase-type approximations. We show that the corrected phase-type approximations exhibit a nice behavior both in finite and infinite time horizon, and we check their accuracy through numerical experiments.     

Solving the Dym initial value problem in reproducing kernel space

We consider two numerical solution approaches for the Dym initial value problem using the reproducing kernel Hilbert space method. For each solution approach, the solution is represented in the form of a series contained in the reproducing kernel space, and a truncated approximate solution is obtained. This approximation converges to the exact solution of the Dym problem when a sufficient number of terms are included. In the first approach, we avoid to perform the Gram-Schmidt orthogonalization process on the basis functions, and this will decrease the computational time. Meanwhile, in the second approach, working with orthonormal basis elements gives some numerical advantages, despite the increased computational time. The latter approach also permits a more straightforward convergence analysis. Therefore, there are benefits to both approaches. After developing the reproducing kernel Hilbert space method for the numerical solution of the Dym equation, we present several numerical experiments in order to show that the method is efficient and can provide accurate approximations to the Dym initial value problem for sufficiently regular initial data after relatively few iterations. We present the absolute error of the results when exact solutions are known and residual errors for other cases. The results suggest that numerically solving the Dym initial value problem in reproducing kernel space is a useful approach for obtaining accurate solutions in an efficient manner.     

Experiments on orthogonalization by biorthogonal representations of orthogonal projectors

A number of experiments are performed with the aim of enhancing a particular feature arising when biorthogonal sequences are used for the purpose of orthogonalization. It is shown that an orthogonalization process executed by biorthogonal sequences and followed by a re-orthogonalization step admits four numerically different realizations. The four possibilities are originated by the fact that, although an orthogonal projector is by definition a self-adjoint operator, due to numerical errors in finite precision arithmetic the biorthogonal representation does not fulfil such a property. In the experiments presented here one of the realizations is shown clearly numerically superior to the remaining three.     

General Full Implicit Strong Taylor Approximations for Stiff Stochastic Differential Equations

In this paper, we present the backward stochastic Taylor expansions for a Ito process, including backward Ito-Taylor expansions and backward Stratonovich-Taylor expansions. We construct the general full implicit strong Taylor approximations (including Ito-Taylor and Stratonovich-Taylor schemes) with implicitness in both the deterministic and the stochastic terms for the stiff stochastic differential equations (SSDE) by employing truncations of backward stochastic Taylor expansions. We demonstrate that these schemes will converge strongly with corresponding order $1,2,3,\ldots$ Mean-square stability has been investigated for full implicit strong Stratonovich-Taylor scheme with order $2$, and it has larger mean-square stability region than the explicit and the semi-implicit strong Stratonovich-Taylor schemes with order $2$. We can improve the stability of simulations considerably without too much additional computational effort by using our full implicit schemes. The full implicit strong Taylor schemes allow a larger range of time step sizes than other schemes and are suitable for SSDE with stiffness on both the drift and the diffusion terms. Our numerical experiment shows these points.     

A space‐time continuous Galerkin method with mesh modification for viscoelastic wave equations

In this article, we consider the space‐time continuous Galerkin (STCG) method for the viscoelastic wave equations. It allows variable temporal step‐sizes, and the changing of the spatial grids in two adjacent time levels. The existence, uniqueness, and stability of the approximate solutions are demonstrated and the error estimates with global and local spatial mesh sizes in norm are derived without any restrictive assumptions on the space‐time meshes. If the meshes in each time level satisfy some reasonable assumptions, then we can get the optimal order error estimates both in time and space. Finally, we give a numerical example on unstructured meshes to confirm the theoretical findings. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1183–1207, 2017     

Reproducing kernel method to solve parabolic partial differential equations with nonlocal conditions

In this study, the parabolic partial differential equations with nonlocal conditions are solved. To this end, we use the reproducing kernel method (RKM) that is obtained from the combining fundamental concepts of the Galerkin method, and the complete system of reproducing kernel Hilbert space that was first introduced by Wang et al. who implemented RKM without Gram–Schmidt orthogonalization process. In this method, first the reproducing kernel spaces and their kernels such that satisfy the nonlocal conditions are constructed, and then the RKM without Gram–Schmidt orthogonalization process on the considered problem is implemented. Moreover, convergence theorem, error analysis theorems, and stability theorem are provided in detail. To show the high accuracy of the present method several numerical examples are solved.     

CONSTRAINT-PRESERVING ENERGY-STABLE SCHEME FOR THE 2D SIMPLIFIED ERICKSEN-LESLIE SYSTEM

Here we consider the numerical approximations of the 2D simplified Ericksen-Leslie system.We first rewrite the system and get a new system.For the new system,we propose an easy-to-implement time discretization scheme which preserves the sphere constraint at each node,enjoys a discrete energy law,and leads to linear and decoupled elliptic equations to be solved at each time step.A discrete maximum principle of the schemc in the finite element form is also proved.Some numerical simulations are performed to validate the scheme and simulate the dynamic motion of liquid crystals.     

Numerical solution of the problem of the stress-strain state in hollow cylinders using spline approximations

The three-dimensional theory of elasticity is used for a study of the stress-strain state in a hollow cylinder with varying stiffness. The corresponding problem is solved by a method that is partly analytical and partly numerical in nature: Spline approximations and collocation are used to reduce the partial differential equations of elasticity to a boundary-value problem for a system of ordinary differential equations of higher order for the radial coordinate, which is then solved using the method of stable discrete orthogonalization. Results for an inhomogeneous cylinder for various types of stiffness are presented.     

Stochastic methods for Dirichlet problems

Using an equivalent expression for solutions of second order Dirichlet problems in terms of Ito type stochastic differential equations, we develop a numerical solution method for Dirichlet boundary value problems. It is possible with this idea to solve for solution values of a partial differential equation at isolated points without having to construct any kind of mesh and without knowing approximations for the solution at any other points. Our method is similar to a recently published approach, but differs primarily in the handling of the boundary. Some numerical examples are presented, applying these techniques to model Laplace and Poisson equations on the unit disk. Visiting Professor, Universidad de Salamanca.     

Reliable preconditioned iterative linear solvers for some numerical integrators

Implicit time‐step numerical integrators for ordinary and evolutionary partial differential equations need, at each step, the solution of linear algebraic equations that are unsymmetric and often large and sparse. Recently, a block preconditioner based on circulant approximations for the linear systems arising in the boundary value methods (BVMs) was introduced by the author. Here, some circulant approximations are compared and a further new type is considered. Numerical experiments are presented to check the effectiveness of the various approximations that can be used in the underlying block preconditioner. Copyright © 2001 John Wiley & Sons, Ltd.     

EXPLICITLY SOLVABLE CHARACTERISTIC METHOD FOR CONVECTION DIFFUSION PROBLEMS

In this paper, a new method for numerically solving nonlinear convection-dominated diffusion problems is devised and analysed. The discrete time approximations with time stepping along charactcristics are cstablished and solved in spaces posscssing reproducing kernel functions. At each time step, the exact solution of the approximate problem is given by explicit expression. The computational advantage of this method is that the schemes are absolutely stable, and are explicitly solvable as well. The stability and error estimates are derived. Some numerical results are given.     

A mixed formulation for the direct approximation of <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-weighted controls for the linear heat equation

This paper deals with the numerical computation of null controls for the linear heat equation. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a given positive time. In [Fernandez-Cara & Münch, Strong convergence approximations of null controls for the 1D heat equation, 2013], a so-called primal method is described leading to a strongly convergent approximation of distributed control: the controls minimize quadratic weighted functionals involving both the control and the state and are obtained by solving the corresponding optimality conditions. In this work, we adapt the method to approximate the control of minimal square integrable-weighted norm. The optimality conditions of the problem are reformulated as a mixed formulation involving both the state and its adjoint. We prove the well-posedeness of the mixed formulation (in particular the inf-sup condition) then discuss several numerical experiments. The approach covers both the boundary and the inner situation and is valid in any dimension.     

Numerical Integration of Reaction–Diffusion Systems

Approximating numerically the solutions of a reaction–diffusion system in an efficient manner requires the application of implicit methods, since the Courant–Friedrichs–Lewy condition on explicit methods imposes a time step of the order of the square of the space step. In this article, we review two types of strategies which are expected to yield reasonably precise solutions within a reasonable computing time. The first examines methods for solving the linear step necessary in any resolution procedure; estimates of CPU time in terms of the error are given in the non preconditioned and in the preconditioned case – provided that it is possible to define an efficient preconditioner. The second strategy is based on splitting, with or without extrapolation. The respective faults and qualities of both strategies are examined; they lead to a list of difficult analytical and numerical problems with possible hints as to their solution.