For a class of non-monotone reaction-diffusion equations with time-delay, the large time-delay usually causes the traveling waves to be oscillatory. In this paper, we are interested in the global stability of these oscillatory traveling waves, in particular, the challenging case of the critical traveling waves with oscillations. We prove that, the critical oscillatory traveling waves are globally stable with the algebraic convergence rate $t$^−1/2, and the non-critical traveling waves are globally stable with the exponential convergence rate $t$^−1/2$e$^−$µt$ for some positive constant $µ$, where the initial perturbations around the oscillatory traveling wave in a weighted Sobolev can be arbitrarily large. The approach adopted is the technical weighted energy method with some new development in establishing the boundedness estimate of the oscillating solutions, which, with the help of optimal decay estimates by deriving the fundamental solutions for the linearized equations, can allow us to prove the global stability and to obtain the optimal convergence rates. Finally, numerical simulations in different cases are carried out, which further confirm our theoretical stability for oscillatory traveling waves, where the initial perturbations can be large.

A linear numerical scheme for an epitaxial growth model is analyzed in this work. The considered scheme is already established in the literature using a convexity splitting argument. We show that it can be naturally derived from an optimization viewpoint using a DC (difference of convex functions) programming framework. Moreover, we prove the convergence of the scheme towards equilibrium by means of the Lojasiewicz-Simon inequality. The fully discrete version, based on a Fourier collocation method, is also analyzed. Finally, numerical simulations are carried out to accommodate our analysis.

Motivated by our recent work about pollution-free difference schemes for solving Helmholtz equation with high wave numbers, this paper presents an analysis of error estimate for the numerical solution on the annulus and hollow sphere domains. By applying the weighted-test-function method and defining two special interpolation operators, we first derive the existence, uniqueness, stability and the pollution-free error estimate for the one-dimensional problems generated from a method based on separation of variables. Utilizing the spherical harmonics and approximations results, we then prove the pollution-free error estimate in $L^2$-norm for multi-dimensional Helmholtz problems.

The sine-Gordon equation is one of the basic equations in modern nonlinear wave theory. It has applications in many areas of physics and mathematics. In this paper, we develop and analyze an energy-conserving local discontinuous Galerkin (LDG) method for the two-dimensional sine-Gordon nonlinear hyperbolic equation on Cartesian grids. We prove the energy conserving property, the $L^2$ stability, and optimal $L^2$ error estimates for the semi-discrete method. More precisely, we identify special numerical fluxes and a suitable projection of the initial conditions for the LDG scheme to achieve $p$ + 1 order of convergence for both the potential and its gradient in the $L^2$-norm, when tensor product polynomials of degree at most $p$ are used. We present several numerical examples to validate the theoretical results. Our numerical examples show the sharpness of the $\mathcal{O}$($h^{p+1}$) estimate.

A decoupled finite element algorithm with different time steps on different physical variables for a Stokes-Darcy interface system coupled with the solution transport is studied. The viscosity of the Stokes equation is assumed to depend on the concentration of the transported solution. The numerical algorithm consists of two steps. In the first step, the system is decoupled on the interface. In the second step, the time derivatives are discretized with different step sizes for different partial differential equations in the system. A careful error analysis provides a guidance on the ratio of the step sizes with respect to the ratio of the physical parameters. Numerical examples are presented to verify the theoretical results and illustrate the effectiveness of the decoupled algorithm of using different time steps.

We consider singularly perturbed boundary value problems with a simple interior turning point whose solutions exhibit an interior layer. These problems are discretised using higher order finite elements on layer-adapted graded meshes proposed by Liseikin. We prove $ε$-uniform error estimates in the energy norm. Furthermore, for linear elements we are able to prove optimal order $ε$-uniform convergence in the $L$²-norm on these graded meshes.