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Comp."],"abstract":"<p>\n                    In this paper, we consider the linear convection-diffusion equation in one dimension with periodic boundary conditions, and analyze the stability of fully discrete methods that are defined with local discontinuous Galerkin (LDG) methods in space and several implicit-explicit (IMEX) Runge-Kutta methods in time. By using the forward temporal differences and backward temporal differences, respectively, we establish two general frameworks of the energy-method based stability analysis. From here, the fully discrete schemes being considered are shown to have monotonicity stability, i.e. the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L squared\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">L^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    norm of the numerical solution does not increase in time, under the time step condition\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"tau less-than-or-equal-to script upper F left-parenthesis h slash c comma d slash c squared right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c4\n                              \n                            <\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">F<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>h<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>c<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>d<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:msup>\n                              <mml:mi>c<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\tau \\le \\mathcal {F}(h\/c, d\/c^2)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , with the convection coefficient\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"c\">\n                        <mml:semantics>\n                          <mml:mi>c<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">c<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , the diffusion coefficient\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"d\">\n                        <mml:semantics>\n                          <mml:mi>d<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">d<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and the mesh size\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"h\">\n                        <mml:semantics>\n                          <mml:mi>h<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">h<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The function\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper F\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">F<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathcal {F}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    depends on the specific IMEX temporal method, the polynomial degree\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of the discrete space, and the mesh regularity parameter. Moreover, the time step condition becomes\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"tau less-than-or-equivalent-to h slash c\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c4\n                              \n                            <\/mml:mi>\n                            <mml:mo>\n                              \u2272\n                              \n                            <\/mml:mo>\n                            <mml:mi>h<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>c<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\tau \\lesssim h\/c<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in the convection-dominated regime and it becomes\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"tau less-than-or-equivalent-to d slash c squared\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c4\n                              \n                            <\/mml:mi>\n                            <mml:mo>\n                              \u2272\n                              \n                            <\/mml:mo>\n                            <mml:mi>d<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:msup>\n                              <mml:mi>c<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\tau \\lesssim d\/c^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in the diffusion-dominated regime. The result is improved for a first order IMEX-LDG method. To complement the theoretical analysis, numerical experiments are further carried out, leading to slightly stricter time step conditions that can be used by practitioners. Uniform stability with respect to the strength of the convection and diffusion effects can especially be relevant to guide the choice of time step sizes in practice, e.g. when the convection-diffusion equations are convection-dominated in some sub-regions.\n                  <\/p>","DOI":"10.1090\/mcom\/3842","type":"journal-article","created":{"date-parts":[[2023,3,15]],"date-time":"2023-03-15T09:38:35Z","timestamp":1678873115000},"page":"2475-2513","source":"Crossref","is-referenced-by-count":4,"title":["Uniform stability for local discontinuous Galerkin methods with implicit-explicit Runge-Kutta time discretizations for linear convection-diffusion equation"],"prefix":"10.1090","volume":"92","author":[{"given":"Haijin","family":"Wang","sequence":"first","affiliation":[]},{"given":"Fengyan","family":"Li","sequence":"additional","affiliation":[]},{"given":"Chi-Wang","family":"Shu","sequence":"additional","affiliation":[]},{"given":"Qiang","family":"Zhang","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2023,5,15]]},"reference":[{"key":"1","series-title":"Pure and Applied Mathematics, Vol. 65","volume-title":"Sobolev spaces","author":"Adams, Robert A.","year":"1975"},{"issue":"2-3","key":"2","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1016\/S0168-9274(97)00056-1","article-title":"Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations","volume":"25","author":"Ascher, Uri M.","year":"1997","journal-title":"Appl. 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