Stability and Error Estimates of Local Discontinuous Galerkin Methods with Implicit–Explicit Backward Difference Formulas up to Fifth Order for Convection–Diffusion Equation

Abstract

In this paper, the stability analysis and optimal error estimates are presented for a kind of fully discrete schemes for solving one-dimensional linear convection–diffusion equation with periodic boundary conditions. The fully discrete schemes are defined with local discontinuous Galerkin (LDG) spatial discretization methods coupled with implicit–explicit (IMEX) temporal discretization methods based on backward difference formulas (BDF). By combining an improved multiplier technique used in the stability analysis for multistep methods and the technique to deal with derivative and jump in LDG methods, we establish a general framework of stability analysis for the corresponding fully discrete LDG–IMEX–BDF schemes up to fifth order in time. The considered schemes are proved to be unconditionally stable, in the sense that a properly defined “discrete energy” is dissipative if the time step is upper bounded by a constant which is independent of the mesh size. Optimal orders of the $L^2$ norm accuracy in both space and time are also proved by energy analysis. Numerical tests are presented to validate the theoretical results.

Appendix

1.1 A.1: The coefficients in Lemmas 3.1 and 4.2 for $s=1$ and $s=2$

For $s=1$ and $s=2$, the coefficients of $\kappa _1$, $\kappa _2$, $\{d_i\}_{i=1}^{s-1}$ and the entries of matrix ${\mathbb {G}}$ are well known, see [16]. Specifically, for $s=1$, we have

$$\begin{aligned}&\kappa _1 = \frac{1}{2}, \quad \kappa _2 = 1, \quad g_{11}=\frac{1}{2}, \end{aligned}$$

(A.1)

and for $s=2$, we have

$$\begin{aligned}&\kappa _1 = \frac{1}{6}, \quad \kappa _2 = \frac{3}{2}, \quad d_1=0, \nonumber \\&g_{11}=\frac{1}{6}, \quad g_{12}=-\frac{1}{3}, \quad g_{22}=\frac{5}{6}. \end{aligned}$$

(A.2)

Obviously, $\kappa _1>0$ and $\kappa _2>0$. The positive-definiteness of ${\mathbb {G}}$ can be checked easily. Furthermore, in (3.2) we can simply take $\lambda _0 = \beta$ for these two cases, thus $A=0$ and $\{c_i\}_{i=1}^{s}=0$.

Similarly, in (4.13) we can take ${\tilde{\lambda }}_0=\varepsilon _0=\frac{\beta }{2}$ for these two cases, so $B=0$ and $\{e_i\}_{i=1}^{s}=0$.

1.2 A.2: The coefficients in Lemmas 3.1 and 4.2 for $s=3$

For $s=3$, $\kappa _1$, $\kappa _2$, $\{d_i\}_{i=1}^2$ and the entries of matrix ${\mathbb {G}}$ are the same as that presented in [17], namely,

$$\begin{aligned}&\kappa _1 = \frac{8-\sqrt{30}}{22}, \quad \kappa _2 = \frac{11(8+\sqrt{30})}{102}, \quad d_1 = \frac{\sqrt{30}-9}{17}, \quad d_2 = \frac{2(9-\sqrt{30})}{17}, \nonumber \\&g_{11}= \frac{8+\sqrt{30}}{187}, \quad g_{12}=\frac{-3(8+\sqrt{30})}{187}, \quad g_{13}=\frac{3(8+\sqrt{30})}{187}, \nonumber \\&g_{22}=\frac{190+11\sqrt{30}}{374}, \quad g_{23}=-\frac{99+6 \sqrt{30}}{187}, \quad g_{33}=\frac{14+\sqrt{30}}{22}. \end{aligned}$$

(A.3)

Obviously, $\kappa _1>0$ and $\kappa _2>0$. The positive-definiteness of ${\mathbb {G}}$ can be verified by checking its eigenvalues. In fact, the eigenvalues of ${\mathbb {G}}$ are approximately equal to 0.0010644088, 0.07207072500, 1.553380287.

Taking $\lambda _0=\frac{3}{11}$, we obtain one of the solutions for $\{c_i\}_{i=1}^3$ and the entries of matrix ${\mathbb {A}}$, they are given as

$$\begin{aligned}&a_{11}= \frac{333-54\sqrt{30}}{3179}+ z_*, \quad a_{12}=0, \quad a_{22}=\frac{3}{11}, \nonumber \\&c_1 = \sqrt{z_*}, \quad c_2=-\frac{\sqrt{z_*}}{18}(66\sqrt{30}z_*+ 407 z_* - 18\sqrt{30}-66), \quad c_3 = 0, \end{aligned}$$

(A.4)

where $z_*$ is one of the real root of $10106041 z^2+(2536842-858330\sqrt{30})z -143856\sqrt{30} + 793476$. $z_*= \frac{135 \sqrt{30}- 399 \pm 3\sqrt{-9725 + 4014\sqrt{30}}}{3179}$, which are approximately equal to 0.2115784868 and 0.0025929210, both values give positive $a_{11}$, and thus the matrix ${\mathbb {A}}$ is positive-definite. Taking $z_*\approx 0.0025929210$, we get the eigenvalues of ${\mathbb {A}}$ are approximately 0.0143040939 and 0.2727272727.

Further taking ${\tilde{\lambda }}_0=\frac{3}{11}$ and $\varepsilon _0=\frac{1}{11}$, we obtain one of the solutions for $\{e_i\}_{i=1}^3$ and the entries of matrix ${\mathbb {B}}$, they are given as

$$\begin{aligned}&b_{11}= \frac{333-54\sqrt{30}}{3179}+ {{\tilde{z}}}_*, \quad b_{12}=0, \quad b_{22}=\frac{2}{11}, \nonumber \\&e_1 = \sqrt{{\tilde{z}}_*}, \quad e_2=-\frac{\sqrt{{\tilde{z}}_*}}{18}(66\sqrt{30}{\tilde{z}}_*+ 407 {\tilde{z}}_* - 12\sqrt{30}-29), \quad e_3 = 0, \end{aligned}$$

(A.5)

where ${\tilde{z}}_*$ is one of the real root of $10106041 z^2+(3455573-858330\sqrt{30})z -143856\sqrt{30} + 793476$. ${\tilde{z}}_*= \frac{270 \sqrt{30}- 1087 \pm \sqrt{194665 - 11556\sqrt{30}}}{6358}$, which are approximately equal to 0.1186381176 and 0.00462419918, both values give positive $b_{11}$, and thus the matrix ${\mathbb {B}}$ is positive-definite. Taking ${\tilde{z}}_*\approx 0.00462419918$, we get the eigenvalues of ${\mathbb {B}}$ are approximately 0.01633537215 and 0.1818181818.

1.3 A.3: The coefficients in Lemmas 3.1 and 4.2 for $s=4$

For $s=4$, $\kappa _1$, $\kappa _2$, $\{d_i\}_{i=1}^3$ and the entries of matrix ${\mathbb {G}}$ are given as follows, in the sense that (3.1) is satisfied with tolerance error of $10^{-31}$ for each coefficient of $w_iw_j$.

$$\begin{aligned}&\kappa _1 = 0.094405276410813782029324113702474\\&\kappa _2 = 1.3240785340858522537272990279540\\&d_1 = 0.1788333104652450268422043512181\\&d_2 = -0.7221089245692809718953051690182\\&d_3 = 0.9077179177428268632639972843823\\&g_{11} = 0.038133461781672544907346212005075\\&g_{12} = -0.15253384712669017962938484802030\\&g_{13} = 0.22880077069003526944407727203044\\&g_{14} = -0.15253384712669017962938484802030\\&g_{22} = 0.62267472907752741152854684200519\\&g_{23} = -0.95777945055888490696888889092537\\&g_{24} = 0.65288356966196203633489381562917\\&g_{33} = 1.5247877374865469936834570807201\\&g_{34} = -1.0881655979438535337816330871974\\&g_{44} = 0.90559472358918621797067588629753 \end{aligned}$$

We observe that $\kappa _1 >0$ and $\kappa _2 >0$, and we can get that the eigenvalues of ${\mathbb {G}}$ are 0.00004826514323140203, 0.009410411961448486, 0.11980927324105947, 2.9619227015891934, thus ${\mathbb {G}}$ is positive-definite.

Taking $\lambda _0=\frac{3}{25}$, we obtain one of the solutions for $\{c_i\}_{i=1}^4$ and the entries of matrix ${\mathbb {A}}$, they are given as follows in the sense that (3.2) is satisfied with tolerance error of $10^{-33}$ for each coefficient of $w_iw_j$.

$$\begin{aligned}&a_{11} = 0.16874145202317428717962251016930\\&a_{12} = -0.019845053879763579080902610935712\\&a_{13} = 0.086087159363004883069812195584665\\&a_{22} = 0.23142908275626820826650337022511\\&a_{23} = -0.10025807527620447894395870706770\\&a_{33} = 0.35720726999415194238802972205615\\&c_1 = -0.40608335310788010829308087860211\\&c_2 = 0.010708634832764590606792808099215\\&c_3 = -0.16402429409974281810379030668411\\&c_4 = 0.052846286585228082721635056225196 \end{aligned}$$

The eigenvalues of ${\mathbb {A}}$ are 0.1302686625796966, 0.18757782163771705, 0.43953132055618094, hence ${\mathbb {A}}$ is also positive-definite.

Further taking ${\tilde{\lambda }}_0=\frac{3}{25}$ and $\varepsilon _0=\frac{1}{25}$, we obtain one of the solutions for $\{e_i\}_{i=1}^4$ and the entries of matrix ${\mathbb {B}}$, they are given as follows in the sense that (4.13) is satisfied with tolerance error of $10^{-33}$ for each coefficient of $w_iw_j$.

$$\begin{aligned}&b_{11} = 0.113402846160428601500976364823521\\&b_{12} = -0.0165239914218789205617197581312801\\&b_{13} = 0.0864518123315534457765888263214463\\&b_{22} = 0.175985438588842511341998542577029\\&b_{23} = -0.0958086228353943283283229290792239\\&b_{33} = 0.315796731347143965821056177851243\\&e_1 = -0.331006168837672201363045348522034\\&e_2 = 0.00310428015834393812757588168298034\\&e_3 = -0.202329124387312680846574633560170\\&e_4 = 0.0648326202837432306332337118149177 \end{aligned}$$

The eigenvalues of ${\mathbb {B}}$ are 0.07674019909822653, 0.13793872805414528, 0.39050608894404326, hence ${\mathbb {B}}$ is also positive-definite.

1.4 A.4: The coefficients in Lemmas 3.1 and 4.2 for $s=5$

For $s=5$, $\kappa _1$, $\kappa _2$, $\{d_i\}_{i=1}^4$ and the entries of matrix ${\mathbb {G}}$ are given as follows, in the sense that (3.1) is satisfied with tolerance error of $10^{-31}$ for each coefficient of $w_iw_j$.

$$\begin{aligned}&\kappa _1 = 0.350177987627029002819774458510987621\\&\kappa _2 = 0.28556906354293456730190447316720111\\&d_1= 0.18260098800206027626812585365905\\&d_2 = -0.15982188227293688184275281238302\\&d_3= 0.14757497527207844159103282503254\\&d_4 = 0.70457917584130857246451188266945\\&g_{11} = 0.0054773756127367704314926533294364326\\&g_{12} = -0.027386878063683852157463266647182164\\&g_{13} = 0.054773756127367704314926533294364326\\&g_{14} = -0.054773756127367704314926533294364326\\&g_{15} = 0.027386878063683852157463266647182164\\&g_{22} = 0.17408060178620846793179788737881\\&g_{23} = -0.37353865498682686201683189446710\\&g_{24} = 0.466114676193774767084995761543426\\&g_{25} = -0.26802596689634056560783491778750736\\&g_{33} = 0.82408941790528321166241394516586\\&g_{34} =-1.077615474265268098667330873960850\\&g_{35} = 0.6348243267981888404663634154788559\\&g_{44} = 1.5733474142189448993061626183034366\\&g_{45} = -0.98147387875975832843667618031647850\\&g_{55} = 0.649822012372970997180225541489012379 \end{aligned}$$

We can find that both $\kappa _1$ and $\kappa _2$ are positive, and we can compute the eigenvalues of ${\mathbb {G}}$, which are 0.000047412145709790024, 0.0011462051548831534, 0.011159345435408816, 0.1293966749328927, 3.0850671842272526, and thus ${\mathbb {G}}$ is positive-definite.

Taking $\lambda _0 = \frac{30}{137}$, we obtain one of the solutions for $\{c_i\}_{i=1}^5$ and the entries of matrix ${\mathbb {A}}$, they are given as follows in the sense that (3.2) is satisfied with tolerance error of $10^{-33}$ for each coefficient of $w_iw_j$.

$$\begin{aligned} a_{11}&\,= 0.085771374306584311437012570362731\\ a_{12}&\,= -0.010494703062067160549519221488008\\ a_{13}&\, = 0.023185001819520047340208566046117\\ a_{22}&\,= 0.091579392945648481920406469649164\\ a_{23}&\, = -0.016563449365737783316923852665598 \\ a_{33}&\,= 0.10015545428640929157576586071102\\ a_{44}&\,= 0.21897810218978102189781021897810\\ c_1&\,= -0.28012490247831085447651251497856\\ c_2&\,= 0.014651058593540641150834062133580\\ c_3&\, = -0.061701494965124979492323370346330\\ c_4&\, = 0.10057311361216397670097017415792\\ a_{14}&\,=a_{24}=a_{34}= c_5 = 0 \end{aligned}$$

The eigenvalues of ${\mathbb {A}}$ are approximately equal to 0.06856358136371647, 0.08113451794744742, 0.1278081222274782, 0.21897810218978103, so ${\mathbb {A}}$ is also positive-definite.

Further taking ${\tilde{\lambda }}_0 = \frac{30}{137}$ and $\varepsilon _0 = \frac{3}{137}$, we obtain one of the solutions for $\{e_i\}_{i=1}^5$ and the entries of matrix ${\mathbb {B}}$, they are given as follows in the sense that (4.13) is satisfied with tolerance error of $10^{-33}$ for each coefficient of $w_iw_j$.

$$\begin{aligned} b_{11}&\,= 0.052213870622550388996794908273166\\ b_{12}&\,= -0.0058547324849932415148953372950837\\ b_{13}&\, =0.024994632905722940377285214793884\\ b_{22}&\,= 0.057813628825767840389204298428498\\ b_{23}&\, = -0.010791681646560130905067278676698 \\ b_{33}&\,= 0.070699996770580707337270093683729\\ b_{44}&\,= 0.19708029197080291970802919708029\\ e_1&\,= -0.21192559379284335099513368195123\\ e_2&\,= -0.0025284544814257540875452061357753\\ e_3&\, = -0.090096509837969863986464582367497\\ e_4&\, = 0.13293832584509150884754980267739\\ b_{14}&\,=b_{24}=b_{34}= e_5 = 0 \end{aligned}$$

The eigenvalues of ${\mathbb {B}}$ are approximately equal to 0.03471657383908016, 0.05359801159997174, 0.09241291077984705, 0.19708029197080293, so ${\mathbb {B}}$ is also positive-definite.