%0 Journal Article %1 citeulike:11422948 %A Acosta, Carlos D. %A Bürger, Raimund %D 2012 %I Oxford University Press %J IMA Journal of Numerical Analysis %K discrete-mollification 65m06-pdes-ibvps-finite-differences 65j20-improperly-posed-problems-regularization %R 10.1093/imanum/drr049 %T Difference schemes stabilized by discrete mollification for degenerate parabolic equations in two space dimensions %U http://dx.doi.org/10.1093/imanum/drr049 %X The discrete mollification method, a convolution-based filtering procedure for the regularization of ill-posed problems, is applied here to stabilize explicit schemes, which were first analysed by Karlsen & Risebro (2001, An operator splitting method for nonlinear convection–diffusion equations. M2AN Math. Model. Numer. Anal. 35, 239–269) for the solution of initial value problems of strongly degenerate parabolic partial differential equations in two space dimensions. Two new schemes are proposed, which are based on directionwise and two-dimensional discrete mollification of the second partial derivatives forming the Laplacian of the diffusion function. The mollified schemes permit substantially larger time steps than the original (basic) scheme. It is proven that both schemes converge to the unique entropy solution of the initial value problem. Numerical examples demonstrate that the mollified schemes are competitive in efficiency, and in many cases significantly more efficient, than the basic scheme.

@article{citeulike:11422948, abstract = {{The discrete mollification method, a convolution-based filtering procedure for the regularization of ill-posed problems, is applied here to stabilize explicit schemes, which were first analysed by Karlsen \& Risebro (2001, An operator splitting method for nonlinear convection–diffusion equations. M2AN Math. Model. Numer. Anal. 35, 239–269) for the solution of initial value problems of strongly degenerate parabolic partial differential equations in two space dimensions. Two new schemes are proposed, which are based on directionwise and two-dimensional discrete mollification of the second partial derivatives forming the Laplacian of the diffusion function. The mollified schemes permit substantially larger time steps than the original (basic) scheme. It is proven that both schemes converge to the unique entropy solution of the initial value problem. Numerical examples demonstrate that the mollified schemes are competitive in efficiency, and in many cases significantly more efficient, than the basic scheme.}}, added-at = {2017-06-29T07:13:07.000+0200}, author = {Acosta, Carlos D. and B\"{u}rger, Raimund}, biburl = {https://www.bibsonomy.org/bibtex/2f430788a221d50cad9e38d353fdc4bc5/gdmcbain}, citeulike-article-id = {11422948}, citeulike-attachment-1 = {acosta_12_difference.pdf; /pdf/user/gdmcbain/article/11422948/837968/acosta_12_difference.pdf; 5034151ab99e1d013a87aee7dda4a49a61d3f678}, citeulike-linkout-0 = {http://dx.doi.org/10.1093/imanum/drr049}, citeulike-linkout-1 = {http://imajna.oxfordjournals.org/content/early/2012/03/16/imanum.drr049.abstract}, citeulike-linkout-2 = {http://imajna.oxfordjournals.org/content/early/2012/03/16/imanum.drr049.full.pdf}, comment = {(private-note)just found this Googling "discrete mollification"}, day = 16, doi = {10.1093/imanum/drr049}, file = {acosta_12_difference.pdf}, interhash = {ad608dc40ae32f7e684e4c0bd9c7c4de}, intrahash = {f430788a221d50cad9e38d353fdc4bc5}, issn = {1464-3642}, journal = {IMA Journal of Numerical Analysis}, keywords = {discrete-mollification 65m06-pdes-ibvps-finite-differences 65j20-improperly-posed-problems-regularization}, month = mar, posted-at = {2012-10-09 13:01:22}, priority = {2}, publisher = {Oxford University Press}, timestamp = {2020-06-30T01:51:32.000+0200}, title = {{Difference schemes stabilized by discrete mollification for degenerate parabolic equations in two space dimensions}}, url = {http://dx.doi.org/10.1093/imanum/drr049}, year = 2012 }