Precise computations of chemotactic collapse using moving mesh methods
Journal of Computational Physics
Applied Numerical Mathematics - Selected papers from the 16th Chemnitz finite element symposium 2003
Numeric vs. symbolic homotopy algorithms in polynomial system solving: a case study
Journal of Complexity - Festschrift for the 70th birthday of Arnold Schönhage
A splitting moving mesh method for reaction-diffusion equations of quenching type
Journal of Computational Physics
Numerical solution of quenching problems using mesh-dependent variable temporal steps
Applied Numerical Mathematics
Discretization of the Cauchy problem for a fast diffusion equation
Journal of Computational and Applied Mathematics
A moving mesh method with variable mesh relaxation time
Applied Numerical Mathematics
Journal of Computational Physics
A grid redistribution method for singular problems
Journal of Computational Physics
Numerical simulation of blowup in nonlocal reaction-diffusion equations using a moving mesh method
Journal of Computational and Applied Mathematics
Moving mesh methods for blowup in reaction-diffusion equations with traveling heat source
Journal of Computational Physics
Applied Numerical Mathematics - Selected papers from the 16th Chemnitz finite element symposium 2003
Numeric vs. symbolic homotopy algorithms in polynomial system solving: a case study
Journal of Complexity - Festschrift for the 70th birthday of Arnold Schönhage
A numerical investigation of blow-up in reaction-diffusion problems with traveling heat sources
Journal of Computational and Applied Mathematics
Computational Solution of Blow-Up Problems for Semilinear Parabolic PDEs on Unbounded Domains
SIAM Journal on Scientific Computing
Optimal mass transport for higher dimensional adaptive grid generation
Journal of Computational Physics
Numerical Blow-up of Semilinear Parabolic PDEs on Unbounded Domains in R2
Journal of Scientific Computing
Moving mesh method for problems with blow-up on unbounded domains
Numerical Algorithms
Journal of Computational and Applied Mathematics
Journal of Computational Physics
An adaptive observation site selection strategy for road traffic data assimilation
Proceedings of the 5th ACM SIGSPATIAL International Workshop on Computational Transportation Science
Convergence analysis of moving finite element methods for space fractional differential equations
Journal of Computational and Applied Mathematics
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In this paper we consider the numerical solution of PDEs with blow-up for which scaling invariance plays a natural role in describing the underlying solution structures. It is a challenging numerical problem to capture the qualitative behaviour in the blow-up region, and the use of nonuniform meshes is essential. We consider moving mesh methods for which the mesh is determined using so-called moving mesh partial differential equations (MMPDEs). Specifically, the underlying PDE and the MMPDE are solved for the blow-up solution and the computational mesh simultaneously. Motivated by the desire for the MMPDE to preserve the scaling invariance of the underlying problem, we study the effect of different choices of MMPDEs and monitor functions. It is shown that for suitable ones the MMPDE solution evolves towards a (moving) mesh which close to the blow-up point automatically places the mesh points in such a manner that the ignition kernel, which is well known to be a natural coordinate in describing the behaviour of blow-up, approaches a constant as $t\to T$ (the blow-up time). Several numerical examples are given to verify the theory for these MMPDE methods and to illustrate their efficacy.