An unconditional stable compact fourth-order finite difference scheme for three dimensional Allen–Cahn equation

In this paper, we present an unconditional stable linear high-order finite difference scheme for three dimensional Allen–Cahn equation. This scheme, which is based on a backward differentiation scheme combined with a fourth-order compact finite difference formula, is second order accurate in time and fourth order accurate in space. A linearly stabilized splitting scheme is used to remove the restriction of time step. We prove the unconditional stability of our proposed method in analysis. A fast and efficient linear multigrid solver is employed to solve the resulting discrete system. We perform various numerical experiments to confirm the high-order accuracy, unconditional stability and efficiency of our proposed method. In particular, we show two applications of our proposed method: triply-periodic minimal surface and volume inpainting.     

A two-level finite element method for the Allen–Cahn equation

We consider the fully implicit treatment for the nonlinear term of the Allen–Cahn equation. To solve the nonlinear problem efficiently, the two-level scheme is employed. We obtain the discrete energy law of the fully implicit scheme and two-level scheme with finite element method. Also, the convergence of the two-level method is presented. Finally, some numerical experiments are provided to confirm the theoretical analysis.     

Numerical solutions of the Burgers–Huxley equation by the IDQ method

In this paper, the Burgers–Huxley equation has been solved by a generalized version of the Iterative Differential Quadrature (IDQ) method for the first time. The IDQ method is a method based on the quadrature rules. It has been proposed by the author applying to a certain class of non-linear problems. Stability and error analysis are performed, showing the efficiency of the method. Besides, an error bound is tried. In the discussion about the numerical examples, the generalized Burgers–Huxley equation is involved too.     

Numerical Analysis of Fully Discretized Crank–Nicolson Scheme for Fractional-in-Space Allen–Cahn Equations

We consider numerical methods for solving the fractional-in-space Allen–Cahn equation which contains small perturbation parameters and strong nonlinearity. A standard fully discretized scheme for this equation is considered, namely, using the conventional second-order Crank–Nicolson scheme in time and the second-order central difference approach in space. For the resulting nonlinear scheme, we propose a nonlinear iteration algorithm, whose unique solvability and convergence can be proved. The nonlinear iteration can avoid inverting a dense matrix with only \(\mathcal {O}(N\log N)\) computation complexity. One major contribution of this work is to show that the numerical solutions satisfy discrete maximum principle under reasonable time step constraint. Based on the maximum stability, the nonlinear energy stability for the fully discrete scheme is established, and the corresponding error estimates are investigated. Numerical experiments are performed to verify the theoretical results.     

Numerical solution of a coupled modified Korteweg–de Vries equation by the Galerkin method using quadratic B-splines

In this paper, numerical solutions of a coupled modified Korteweg–de Vries equation have been obtained by the quadratic B-spline Galerkin finite element method. The accuracy of the method has been demonstrated by five test problems. The obtained numerical results are found to be in good agreement with the exact solutions. A Fourier stability analysis of the method is also investigated.     

Comparison study on the different dynamics between the Allen–Cahn and the Cahn–Hilliard equations

We perform a comparison study on the different dynamics between the Allen–Cahn (AC) and the Cahn–Hilliard (CH) equations. The AC equation describes the evolution of a non-conserved order field during anti-phase domain coarsening. The CH equation describes the process of phase separation of a conserved order field. The AC and the CH equations are second-order and fourth-order nonlinear parabolic partial differential equations, respectively. Linear stability analysis shows that growing and decaying modes for both the equations are the same. While the growth rates are monotonically decreasing with respect to the modes for the AC equation, the growth rates for the CH equation are increasing and then decreasing with respect to the modes. We perform various numerical tests using the Fourier spectral method to highlight the different evolutionary dynamics between the AC and the CH equations.     

Distributed and boundary optimal control of the Allen–Cahn equation with regular potential and dynamic boundary conditions

This paper is concerned with an optimal control problem (P) (both distributed control as well as boundary control) for the nonlinear phase-field (Allen–Cahn) equation, involving a regular potential and dynamic boundary condition. A family of approximate optimal control problems (P^?) is introduced and results for the existence of an optimal control for problems (P) and (P^?) are proven. Furthermore, the convergence result of the optimal solution of problem (P^?) to the optimal solution of problem (P) is proved. Besides the existence of an optimal control in problem (P^?), necessary optimality conditions (Pontryagin's principle) as well as a conceptual gradient-type algorithm to approximate the optimal control, were established in the end.     

Solitary-wave solutions of the Degasperis–Procesi equation by means of the homotopy analysis method

The homotopy analysis method (HAM) is applied to the Degasperis–Procesi equation in order to find analytic approximations to the known exact solitary-wave solutions for the solitary peakon wave and the family of solitary smooth-hump waves. It is demonstrated that the approximate solutions agree well with the exact solutions. This provides further evidence that the HAM is a powerful tool for finding excellent approximations to nonlinear solitary waves.     

A high-order compact method for nonlinear Black–Scholes option pricing equations of American options

Due to transaction costs, illiquid markets, large investors or risks from an unprotected portfolio the assumptions in the classical Black–Scholes model become unrealistic and the model results in nonlinear, possibly degenerate, parabolic diffusion–convection equations. Since in general, a closed-form solution to the nonlinear Black–Scholes equation for American options does not exist (even in the linear case), these problems have to be solved numerically. We present from the literature different compact finite difference schemes to solve nonlinear Black–Scholes equations for American options with a nonlinear volatility function. As compact schemes cannot be directly applied to American type options, we use a fixed domain transformation proposed by ?ev?ovi? and show how the accuracy of the method can be increased to order four in space and time.     

Finite element approximation of the Cahn–Hilliard equation on surfaces

In this paper, we consider the phase separation on general surfaces by solving the nonlinear Cahn–Hilliard equation using a finite element method. A fully discrete approximation scheme is introduced, and we establish a priori estimates for the discrete solution that does not rely on any knowledge of the exact solution beyond the initial time. This in turn leads to convergence and optimal error estimates of the discretization scheme. Numerical examples are also provided to substantiate the theoretical results.     

Numerical simulation of coupled nonlinear Schrödinger equation

The coupled nonlinear Schrödinger equation models several interesting physical phenomena. It represents a model equation for optical fiber with linear birefringence. In this paper we introduce a finite difference method for a numerical simulation of this equation. This method is second-order in space and conserves the energy exactly. It is quite accurate and describes the interaction picture clearly according to our numerical results.     

A compact ADI Crank–Nicolson difference scheme for the two-dimensional time fractional subdiffusion equation

In this paper, a compact alternating direction implicit (ADI) Crank–Nicolson difference scheme is proposed and analysed for the solution of two-dimensional time fractional subdiffusion equation. The Riemann–Liouville time fractional derivative is approximated by the weighted and shifted Grünwald difference operator and the spatial derivative is discretized by a fourth-order compact finite difference method. The stability and convergence of the difference scheme are discussed and theoretically proven by using the energy method. Finally, numerical experiments are carried out to show that the numerical results are in good agreement with the theoretical analysis.     

Numerical simulation of Richtmyer–Meshkov instability driven by imploding shocks

In this paper, the classical piecewise parabolic method (PPM) is generalized to compressible two-fluid flows, and is applied to simulate Richtmyer–Meshkov instability (RMI) induced by imploding shocks. We use the compressible Euler equations together with an advection equation for volume fraction of one fluid component as model system, which is valid for both pure fluid and two-component mixture. The Lagrangian-remapping version of PPM is employed to solve the governing equations with dimensional-splitting technique incorporated for multi-dimensional implementation, and the scheme proves to be non-oscillatory near material interfaces. We simulate RMI driven by imploding shocks, examining cases of single-mode and random-mode perturbations on the interfaces and comparing results of this instability in planar and cylindrical geometries. Effects of perturbation amplitude and shock strength are also studied.     

Pseudospectral method of solution of the Fitzhugh–Nagumo equation

We present a study of the convergence of different numerical schemes in the solution of the Fitzhugh–Nagumo equations in the form of two coupled reaction diffusion equations for activator and inhibitor variables. The diffusion coefficient for the inhibitor is taken to be zero. The Fitzhugh–Nagumo equations, have spatial and temporal dynamics in two different scales and the solutions exhibit shock-like waves. The numerical schemes employed are a Chebyshev multidomain method, a finite difference method and the method developed by Barkley [D. Barkley, A model for fast computer simulation of excitable media, Physica D, 49 (1991) 61–70]. We consider two different models for the local dynamics. We present results for plane wave propagation in one dimension and spiral waves for two dimensions. We use an operator splitting method with the Chebyshev multidomain approach in order to reduce the computational time. Zero flux boundary conditions are imposed on the solutions.     

Energy stable compact scheme for Cahn–Hilliard equation with periodic boundary condition

We present a compact scheme to solve the Cahn–Hilliard equation with a periodic boundary condition, which is fourth-order accurate in space. We introduce schemes for two and three dimensions, which are derived from the one-dimensional compact stencil. The energy stability is completely proven for the proposed scheme based on the application of the compact method and well-known convex splitting methods. Detailed proofs of the mass conservation and unique solvability are also established. Numerical experiments are presented to demonstrate the accuracy and stability of the proposed methods.     

Implementation and solution of the diffusion–reaction equation using high-order methods

The orthogonal collocation, Galerkin, tau and least-squares methods are applied to solve a diffusion–reaction problem. In general, the least-squares method suffers from lower accuracy than the other weighted residual methods. The least-squares method holds the most complex linear algebra theory and is thus associated with the most complex implementation. On the other hand, an advantage of the least-squares method is that it always produces a symmetric and positive definite system matrix which can be solved with an efficient iterative technique such as the conjugate gradient method or its preconditioned version. For the present problem, neither the Galerkin, tau and orthogonal collocation techniques produce symmetric and positive definite system matrices, hence the conjugate gradient method is not applicable for these numerical techniques.     

Fast local image inpainting based on the Allen–Cahn model

In this paper, we propose a fast local image inpainting algorithm based on the Allen–Cahn model. The proposed algorithm is applied only on the inpainting domain and has two features. The first feature is that the pixel values in the inpainting domain are obtained by curvature-driven diffusions and utilizing the image information from the outside of the inpainting region. The second feature is that the pixel values outside of the inpainting region are the same as those in the original input image since we do not compute the outside of the inpainting region. Thus the proposed method is computationally efficient. We split the governing equation into one linear equation and one nonlinear equation by using an operator splitting technique. The linear equation is discretized by using a fully implicit scheme and the nonlinear equation is solved analytically. We prove the unconditional stability of the proposed scheme. To demonstrate the robustness and accuracy of the proposed method, various numerical results on real and synthetic images are presented.     

Numerical Approximations for the Cahn–Hilliard Phase Field Model of the Binary Fluid-Surfactant System

In this paper, we consider the numerical approximations for the commonly used binary fluid-surfactant phase field model that consists two nonlinearly coupled Cahn–Hilliard equations. The main challenge in solving the system numerically is how to develop easy-to-implement time stepping schemes while preserving the unconditional energy stability. We solve this issue by developing two linear and decoupled, first order and a second order time-stepping schemes using the so-called “invariant energy quadratization” approach for the double well potentials and a subtle explicit-implicit technique for the nonlinear coupling potential. Moreover, the resulting linear system is well-posed and the linear operator is symmetric positive definite. We rigorously prove the first order scheme is unconditionally energy stable. Various numerical simulations are presented to demonstrate the stability and the accuracy thereafter.     

Numerical simulation of blow-up solutions for the generalized Davey–Stewartson system

Blow-up solutions for the generalized Davey–Stewartson system are studied numerically by using a split-step Fourier method. The numerical method has spectral-order accuracy in space and first-order accuracy in time. To evaluate the ability of the split-step Fourier method to detect blow-up, numerical simulations are conducted for several test problems, and the numerical results are compared with the analytical results available in the literature. Good agreement between the numerical and analytical results is observed.     

Approximate analytical solutions of fractional Benney–Lin equation by reduced differential transform method and the homotopy perturbation method

In this paper, the approximate analytical solutions of Benney–Lin equation with fractional time derivative are obtained with the help of a general framework of the reduced differential transform method (RDTM) and the homotopy perturbation method (HPM). RDTM technique does not require any discretization, linearization or small perturbations and therefore it reduces significantly the numerical computation. Comparing the methodology (RDTM) with some known technique (HPM) shows that the present approach is effective and powerful. The numerical calculations are carried out when the initial conditions in the form of periodic functions and the results are depicted through graphs. The eight different cases have studied and proved that the method is extremely effective due to its simplistic approach and performance.