A Numerical Study of Blowup in the Harmonic Map Heat Flow Using the MMPDE Moving Mesh Method

The numerical solution of the harmonic heat map flow problems with blowup in finite or infinite time is considered using an adaptive moving mesh method. A properly chosen monitor function is derived so that the moving mesh method can be used to simulate blowup and produce accurate blowup profiles which agree with formal asymptotic analysis. Moreover, the moving mesh method has finite time blowup when the underlying continuous problem does. In situations where the continuous problem has infinite time blowup, the moving mesh method exhibits finite time blowup with a blowup time tending to infinity as the number of mesh points increases. The inadequacy of a uniform mesh solution is clearly demonstrated.     

Solitary Waves in Nonlinear Beam Equations: Stability,Fission and Fusion

We continue work by the second author and co-workers onsolitary wave solutions of nonlinear beam equations and their stabilityand interaction properties. The equations are partial differentialequations that are fourth-order in space and second-order in time.First, we highlight similarities between the intricate structure ofsolitary wave solutions for two different nonlinearities; apiecewise-linear term versus an exponential approximation to thisnonlinearity which was shown in earlier work to possess remarkablystable solitary waves. Second, we compare two different numericalmethods for solving the time dependent problem. One uses a fixed griddiscretization and the other a moving mesh method. We use these methodsto shed light on the nonlinear dynamics of the solitary waves. Earlywork has reported how even quite complex solitary waves appear stable,and that stable waves appear to interact like solitons. Here we show twofurther effects. The first effect is that large complex waves can, as aresult of roundoff error, spontaneously decompose into two simplerwaves, a process we call fission. The second is the fusion of twostable waves into another plus a small amount of radiation.     

Simulating finger phenomena in porous media with a moving finite element method

The non-equilibrium Richards equation is solved using a moving finite element method in this paper. The governing equation is discretized spatially with a standard finite element method, and temporally with second-order Runge–Kutta schemes. A strategy of the mesh movement is based on the work by Li et al. [R.Li, T.Tang, P.W. Zhang, A moving mesh finite element algorithm for singular problems in two and three space dimensions, Journal of Computational Physics, 177 (2002) 365–393]. A Beckett and Mackenzie type monitor function is adopted. To obtain high quality meshes around the wetting front, a smoothing method which is based on the diffusive mechanism is used. With the moving mesh technique, high mesh quality and high numerical accuracy are obtained successfully. The numerical convergence and the advantage of the algorithm are demonstrated by a series of numerical experiments.     

Vertices of planar curves under the action of linear transformations

The number of vertices of a smooth Jordan curve with nowhere vanishing curvature can change under the action of a nonsingular real linear transformation. We examine the bifurcation set in the space of linear transformations for the number of vertices on the image curve, showing that generally there is a codimension-one set of linear transformations making an arbitrary point into a vertex, and obtaining conditions that the point be capable of being transformed into a higher vertex. We demonstrate that there is always an open set of linear transformations such that the image curves have at least six vertices. Received 23 July 1999; revised 17 March 2000.     

Higher Order Nonuniform Grids for Singularly Perturbed Convection-Diffusion-Reaction Problems

Computational Mathematics and Mathematical Physics - In this paper, a higher order nonuniform grid strategy is developed for solving singularly perturbed convection-diffusion-reaction problems with...     

A Numerical Study of Two-Phase Flow Models with Dynamic Capillary Pressure and Hysteresis

Saturation overshoot and pressure overshoot are studied by incorporating dynamic capillary pressure, capillary pressure hysteresis and hysteretic dynamic coefficient with a traditional fractional flow equation in one-dimensional space. Using the method of lines, the discretizations are constructed by applying the Castillo–Grone’s mimetic operators in the space direction and a semi-implicit integrator in the time direction. Convergence tests and conservation properties of the schemes are presented. Computed profiles capture both the saturation overshoot and pressure overshoot phenomena. Comparisons between numerical results and experiments illustrate the effectiveness and different features of the models.     

An asymptotic-numerical approach to the coronal loop problem

The coronal loop problem is characterized by mixed boundary conditions and the loop length condition, which is global. Using singular perturbation methods one can identify and construct two boundary layers at the base of the loop. Extending this to a combined asymptotic-numerical treatment it is possible to construct two static solutions satisfying the same conditions; this unusual feature arises from the presence of the first boundary layer, which corresponds with a steep temperature gradient and an energy balance dominated by conduction and radiation losses.