{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T17:41:07Z","timestamp":1680284467769},"reference-count":42,"publisher":"Walter de Gruyter GmbH","issue":"1","funder":[{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","award":["RGPIN-2019-06855"],"award-info":[{"award-number":["RGPIN-2019-06855"]}],"id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this paper, we analyse full discretizations of an initial boundary value problem (IBVP) related to reaction-diffusion equations.\nTo avoid possible order reduction, the IBVP is first transformed into an IBVP with homogeneous boundary conditions (IBVPHBC) via a lifting of inhomogeneous Dirichlet, Neumann or mixed Dirichlet\u2013Neumann boundary conditions.\nThe IBVPHBC is discretized in time via the deferred correction method for the implicit midpoint rule and leads to a time-stepping scheme of order <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mrow>\n                                 <m:mn>2<\/m:mn>\n                                 <m:mo>\u2062<\/m:mo>\n                                 <m:mi>p<\/m:mi>\n                              <\/m:mrow>\n                              <m:mo>+<\/m:mo>\n                              <m:mn>2<\/m:mn>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2021-0167_ineq_0001.png\" \/>\n                        <jats:tex-math>2p+2<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> of accuracy at the stage <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>p<\/m:mi>\n                              <m:mo>=<\/m:mo>\n                              <m:mrow>\n                                 <m:mn>0<\/m:mn>\n                                 <m:mo>,<\/m:mo>\n                                 <m:mn>1<\/m:mn>\n                                 <m:mo>,<\/m:mo>\n                                 <m:mn>2<\/m:mn>\n                                 <m:mo>,<\/m:mo>\n                                 <m:mi mathvariant=\"normal\">\u2026<\/m:mi>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2021-0167_ineq_0002.png\" \/>\n                        <jats:tex-math>p=0,1,2,\\ldots<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> of the correction.\nEach semi-discretized scheme results in a nonlinear elliptic equation for which the existence of a solution is proven using the Schaefer fixed point theorem.\nThe elliptic equation corresponding to the stage \ud835\udc5d of the correction is discretized by the Galerkin finite element method and gives a full discretization of the IBVPHBC.\nThis fully discretized scheme is unconditionally stable with order <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mrow>\n                                 <m:mn>2<\/m:mn>\n                                 <m:mo>\u2062<\/m:mo>\n                                 <m:mi>p<\/m:mi>\n                              <\/m:mrow>\n                              <m:mo>+<\/m:mo>\n                              <m:mn>2<\/m:mn>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2021-0167_ineq_0001.png\" \/>\n                        <jats:tex-math>2p+2<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> of accuracy in time.\nThe order of accuracy in space is equal to the degree of the finite element used when the family of meshes considered is shape-regular, while an increment of one order is proven for a quasi-uniform family of meshes.\nNumerical tests with a bistable reaction-diffusion equation having a strong stiffness ratio, a Fisher equation, a linear reaction-diffusion equation addressing order reduction and two linear IBVPs in two dimensions are performed and demonstrate the unconditional convergence of the method.\nThe orders 2, 4, 6, 8 and 10 of accuracy in time are achieved.\nExcept for some linear problems, the accuracy of DC methods is better than that of BDF methods of same order.<\/jats:p>","DOI":"10.1515\/cmam-2021-0167","type":"journal-article","created":{"date-parts":[[2022,8,24]],"date-time":"2022-08-24T20:03:58Z","timestamp":1661371438000},"page":"219-250","source":"Crossref","is-referenced-by-count":0,"title":["Arbitrary High-Order Unconditionally Stable Methods for Reaction-Diffusion Equations with inhomogeneous Boundary Condition via Deferred Correction"],"prefix":"10.1515","volume":"23","author":[{"given":"Saint-Cyr Elvi Rodrigue","family":"Koyaguerebo-Im\u00e9","sequence":"first","affiliation":[{"name":"D\u00e9partement de Math\u00e9matiques et Informatique , Facult\u00e9 des Sciences , Universit\u00e9 de Bangui , BP 1450 , Bangui , Central African Republic"}]},{"given":"Yves","family":"Bourgault","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics , University of Ottawa , STEM Complex, 150 Louis-Pasteur Pvt , Ottawa , ON K1N 6N5 , Canada"}]}],"member":"374","published-online":{"date-parts":[[2022,8,25]]},"reference":[{"key":"2023033112515601453_j_cmam-2021-0167_ref_001","doi-asserted-by":"crossref","unstructured":"M. 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