A class of new stable, explicit methods to solve the non‐stationary heat equation

This paper introduces a set of new fully explicit numerical algorithms to solve the spatially discretized heat or diffusion equation. After discretizing the space and the time variables according to conventional finite difference methods, these new methods do not approximate the time derivatives by finite differences, but use a combined two-stage constant-neighbour approximation to decouple the ordinary differential equations and solve them analytically. In the final expression for the new values of the variable, the time step size appears not in polynomial or rational, but in exponential form with negative coefficients, which can guarantee stability. The two-stage scheme contains a free parameter p and we analytically prove that the convergence is second order in the time step size for all values of p and the algorithm is unconditionally stable if p is at least 0.5, not only for the linear heat equation, but for the nonlinear Fisher’s equation as well. We compare the performance of the new methods with analytical and numerical solutions. The results suggest that the new algorithms can be significantly faster than the widely used explicit or implicit methods, particularly in the case of extremely large stiff systems.

In this paper we introduce a new type of explicit numerical algorithm to solve the spatially discretized linear heat or diffusion equation. After discretizing the space variables as in standard finite difference methods, this novel method does not approximate the time derivatives by finite differences, but use three stage constant‐neighbor and linear neighbor approximations to decouple the ordinary differential equations and solve them analytically. In the final expression for the new values of the variable, the time step size appears not in polynomial or rational, but in exponential form with negative coefficients, which can guarantee unconditional stability. The scheme contains a free parameter p. We show that the convergence of the method is third‐order in the time step size regardless of the values of p, and, according to von Neumann stability analysis, the method is stable for a wide range of p. We validate the new method by testing the results in a case where the analytical solution exists, then we demonstrate the competitiveness by comparing its performance with several other numerical solvers.

This paper reports on a novel explicit numerical method for the spatially discretized diffusion or heat equation. After discretizing the space variables as in conventional finite difference methods, this method does not use a finite difference approximation for the time derivatives, it instead combines constant‐neighbor and linear‐neighbor approximations, which decouple the ordinary differential equations, thus they can be solved analytically. In the obtained three‐stage method, the time step size appears in exponential form with negative coefficients in the final expression. This property guarantees unconditional stability, as it is shown using von Neumann stability analysis. It is also proved that the convergence of the method is third order in the time step size. After verification, by solving Fisher's and Huxley's equations, it is demonstrated that it works for nonlinear equations as well. The new algorithm is tested against widely used numerical solvers for cases where the media is strongly inhomogeneous. According to the results, the new method is significantly more effective than the traditional explicit or implicit methods, especially for extremely large stiff systems. It is believed that this new method is unique in the sense that it is the first unconditionally stable explicit method with third‐order convergence.

Exact solutions of linear and non-linear differential-difference heat and diffusion equations with finite relaxation time

The numerous examples of Chapters 2 and 3 demonstrate how the classical method of separation of variables is used to generate solutions of Laplace’s and the heat equation over specified finite domains. Utilizing the modern methods inherent in Green’s functions and Green’s Theorem, solutions of Poisson’s equation over an arbitrary finite domain are established. Can such modern methods be applied to the heat equation? Unlike Laplace’s or Poisson’s equations which are static in time, the heat equation evolves in time. Indeed, the heat equation is the prototype for what are commonly called evolution equations. Nonlinear evolution equations are examined in Chapter 6 and a special nonlinear system is detailed in Chapter 7. For now, the focus will be on the linear heat and wave equations. Instead of Green’s Theorem, one of the most powerful ideas in modern mathematics is applied: The Fourier transform. The program of study for this chapter then is to define the Fourier transform, develop its properties, apply it to the heat and wave equations, and derive analytic solutions.

We formulate and study a one–dimensional single–species diffusive–delay population model. The time delay is the time taken from birth to maturity. Without diffusion, the delay differential model extends the well–known logistic differential equation by allowing delayed constant birth processes and instantaneous quadratically regulated death processes. This delayed model is known to have simple global dynamics similar to that of the logistic equation. Through the use of a sub/supersolution pair method, we show that the diffusive delay model continues to generate simple global dynamics. This has the important biological implication that quadratically regulated death processes dramatically simplify the growth dynamics. We also consider the possibility of travelling wavefront solutions of the scalar equation for the mature population, connecting the zero solution of that equation with the positive steady state. Our main finding here is that our fronts appear to be all monotone, regardless of the size of the delay. This is in sharp contrast to the frequently reported findings that delay causes a loss of monotonicity, with the front developing a prominent hump in some other delay models.

An implicit fourth-order compact finite difference scheme for one-dimensional Burgers’ equation

We introduce a novel explicit and stable numerical algorithm to solve the spatially discretized heat or diffusion equation. We compare the performance of the new method with analytical and numerical solutions. We show that the method is first order in time and can give approximate results for extremely large systems faster than the commonly used explicit or implicit methods.

We consider the linear heat equation with potential in a n‐dimensional thin cilinder Ωε=Ω×(0,ε) where Ω is a bounded open smooth set of $\mathbb{R}^{n-1}$ with n≥2 and ε is a small parameter. We study the null controllability problem when the control acts in a cylindrical region ωε=ω×(0,ε), where ω⊂Ω is an open and non‐empty subset of Ω. We prove that, under appropriate boundary conditions, for a suitable class of potentials the heat equation is uniformly null controllable as ε→0. We also prove the convergence of the controls to a null control for the n−1‐dimensional heat equation in Ω. Similar results are proved for the semilinear heat equation with globally Lipschitz nonlinearities.

In this paper we address the problem of null-controllability of heat equations in two different cases: (a) The semilinear heat equation in bounded domains and (b) The linear heat equation in the half line. Concerning the first problem (a) we show that a number of systems in which blow-up arises may be controlled by means of external forces which are localized in an arbitrarily small open set. In the frame of problem (b) we prove that compactly supported initial data may not be driven to zero if the control is supported in a bounded set. This shows that although the velocity of propagation in the heat equation is infinite, this is not sufficient to guarantee null-controllability properties.We also include a list of open problems.KeywordsHeat EquationMoment ProblemApproximate ControllabilityCarleman EstimateOpen Nonempty SubsetThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The aim of this paper is to systematically construct and test novel odd–even hopscotch-type numerical algorithms solving the diffusion or heat equation. Among the studied explicit two-stage methods some of them are unconditionally stable and have second order convergence rate in time step size, which is proved analytically as well. We apply the best methods to the nonlinear Fisher’s equation to demonstrate that they work also for nonlinear equations. Then, in order to examine the competitiveness of the new algorithms, we test them for the heat equation against widely used numerical solvers in cases where the media are strongly inhomogeneous and thus the coefficients strongly depend on space. The results suggest that the new methods are significantly more effective than the widely used explicit or implicit methods, especially for extremely large stiff systems.

In this paper, two efficient fourth-order compact finite difference algorithms have been developed to solve the one-dimensional Burgers’ equation: u t +u u x =ε u xx . The methods are based on the Hopf–Cole transformation, Richardson's extrapolation, and multilevel grids. In both methods, we first transform the original nonlinear Burgers’ equation into a linear heat equation: w t =ε w xx using the Hopf–Cole transformation, which is given as u=−2ε (w x /w). In the first method, the resulted heat equation is solved by the second-order accurate Crank–Nicholson algorithm while w x is approximated by central finite difference, which is also second-order accurate. Richardson's extrapolation technique is then applied in both time and space to obtain fourth-order accuracy. In the second method, to reduce the cancellation error in approximating w x , we derive the heat equation satisfied by w x , which is then solved by the Crank–Nicholson algorithm. The original Dirichlet boundary condition is transformed into the Robin boundary condition, which is also approximated using second-order central finite difference. Finally, Richardson's extrapolation and multilevel grid techniques are applied in both time and space to obtain fourth-order accuracy. To study the efficiency, accuracy and robustness, we solved two numerical examples and the results are compared with those of two other higher-order methods proposed in W. Liao [An implicit fourth-order compact finite difference scheme for one-dimensional Burgers’ equation, Appl. Math. Comput. 206(2) (2008), pp. 755–764] and I.A. Hassanien, A.A. Salama, and H.A. Hosham [Fourth-order finite difference method for solving Burgers’ equation, Appl. Math. Comput. 170 (2005), pp. 781–800].

The on-line or adaptive identification of parameters in abstract linear and nonlinear infinite-dimensional dynamical systems is considered. An estimator in the form of an infinite-dimensional linear evolution system having the state and parameter estimates as its states is defined. Convergence of the state estimator is established via a Lyapunov estimate. The finite-dimensional notion of a plant being sufficiently rich or persistently excited is extended to infinite dimensions. Convergence of the parameter estimates is established under the additional assumption that the plant is persistently excited. A finite-dimensional approximation theory is developed, and convergence results are established. Numerical results for examples involving the estimation of both constant and functional parameters in one-dimensional linear and nonlinear heat or diffusion equations and the estimation of stiffness and damping parameters in a one-dimensional wave equation with Kelvin--Voigt viscoelastic damping are presented.

Solutions of heat or diffusion equations with the boundary conditions which is a dynamic random field are discussed. This kind of method can be used to obtain the description of heat equations or diffusion equations based on observed physical reality, ie ordinary differential equations, representing heat or diffusion propagation, with a boundary condition that satisfies stochastic differential equations. The heat or diffusion equations obtained from the method are the compared to the heat equation or the stochastic diffusion. The comparison is emphasized on the existence and properties of Green functions.

In this thesis, we develop an explicit multi-rate time stepping method for solving parabolic equations on a one dimensional adaptively refined mesh. Parabolic equations are characterized by their stiffness and as a result are usually solved using implicit time stepping schemes [16]. However, implicit schemes have the disadvantage that they can be expensive in higher dimensions or complicated to implement on adaptive or otherwise non-uniform meshes. Moreover, for coupled systems of parabolic equations, it can be difficult to achieve the expected order of accuracy without using sophisticated operator splitting techniques. For these reasons, we seek to exploit the properties of stabilized explicit methods for parabolic equations. In particular, we use the Runge-Kutta-Chebyshev (RKC) methods, a family of explicit Runge-Kutta methods, with numerical stability regions that extend far into the left half plane [12, 15, 21, 22, 26, 27, 28]. A central goal of this thesis is to use a second order RKC scheme to numerically solve parabolic equations on a one dimensional adaptively refined finite volume mesh. To make our implementation efficient, we design a time stepping algorithm in which time step sizes are chosen to respect the local mesh widths. This time stepping process requires communication between the RKC stages on different refinement levels. By linearly interpolating in time between the stage values, we obtain the ghost cell values for the finite volume scheme on each level. To our knowledge, this approach to adaptively refining in time, commonly referred to as a "multi-rate time stepping" strategy, combined with RKC time stepping method has not been previously implemented. We develop our multi-rate algorithm on a one dimensional statically refined mesh using the second order finite volume scheme to numerically solve the heat or diffusion equation on each grid stored in a hierarchy of meshes. Using the "method of manufactured solutions", we demonstrate that our method is second order accurate, and for our test problem, the multi-rate scheme requires only about 20% of the computational work required by the uniformly refined mesh at the same resolution. The algorithm we develop manages the time stepping between the refinement levels only, and so extends directly to higher dimensional problems. Future work in this direction includes applying the new multi-rate RKC time stepping scheme to biological pattern formations or crystal growth in the 2D ForestClaw code [7] on parallel machines.