document: the convection - diffusion model can be expressed mathematically , which is a semi linear parabolic partial differential equation . 
   specially , we consider an initial value system of _ convection - diffusion _ equation in @xmath3 dimension as : @xmath4,\ ] ] together with the dirichlet boundary conditions : @xmath5,\ ] ] or neumann boundary conditions : @xmath6.\ ] ] where @xmath7 is the boundary of computational domain @xmath8 \times [ c , d]\subset { \mathbb{r}}^2 $ ] , @xmath9 $ ] is time interval , and @xmath10 @xmath11 and @xmath12 are known smooth functions , and @xmath13 denote heat or vorticity . 
   the parameters : @xmath14 and @xmath15 are constant convective velocities while the constants @xmath16 are diffusion coefficients in the direction of @xmath17 and @xmath18 , respectively . 
   the convection - diffusion models have remarkable applications in various branches of science and engineering , for instance , fluid motion , heat transfer , astrophysics , oceanography , meteorology , semiconductors , hydraulics , pollutant and sediment transport , and chemical engineering . specially , in computational hydraulics and fluid dynamics to model convection - diffusion of quantities such as mass , heat , energy , vorticity @xcite . 
   many researchers have paid their attention to develop some schemes which could produce accurate , stable and efficient solutions behavior of convection - diffusion problems , see @xcite and the references therein .    in the last years 
   , the convection - diffusion equation has been solved numerically using various techniques : namely- finite element method @xcite , lattice boltzmann method @xcite , finite - difference scheme and higher - order compact finite difference schemes @xcite . a nine - point high - order compact implicit scheme proposed by noye and tan 
   @xcite is third - order accurate in space and second - order accurate in time , and has a large zone of stability . 
   an extension of higher order compact difference techniques for steady - state @xcite to the time - dependent problems have been presented by spotz and carey @xcite , are fourth - order accurate in space and second or lower order accurate in time but conditionally stable . 
   the fourth - order compact finite difference unconditionally stable scheme due to dehghan and mohebbi @xcite have the accuracy of order @xmath19 . 
   a family of unconditionally stable finite difference schemes presented in @xcite have the accuracy of order @xmath20 . 
   the schemes presented in @xcite are based on high - order compact scheme and weighted time discretization , are second or lower order accurate in time and fourth - order accurate in space . 
   the high - order alternating direction implicit ( adi ) scheme with accuracy of order @xmath21 proposed by karaa and zhang @xcite , is unconditionally stable . a high - order unconditionally stable exponential scheme for unsteady @xmath22d convection - diffusion equation by tian and yua @xcite have the accuracy of order @xmath21 . 
   a rational high - order compact alternating direction implicit ( adi ) method have been developed for solving @xmath3d unsteady convection - diffusion problems @xcite is unconditionally stable and have the accuracy of order @xmath21 . a unconditionally stable fourth - order compact finite difference approximation for discretizing spatial derivatives and the cubic @xmath23- spline collocation method in time , proposed by mohebbi and dehghan @xcite , 
   have the accuracy of order @xmath19 . 
   an unconditionally stable , semi - discrete based on pade approximation , by ding and zhang @xcite , is fourth - order accurate in space and in time both . 
   the most of schemes are based on the two - level finite difference approximations with dirichlet conditions , and very few schemes have been developed to solve the convection - diffusion equation with neumann s boundary conditions , see @xcite and references therein . the fourth - order compact finite difference scheme by cao et al . 
   @xcite is of @xmath24th - order accurate in time and 4th - order in the space . 
   a high - order alternating direction implicit scheme based on fourth - order pade approximation developed by you @xcite is unconditionally stable with the accuracy of order @xmath21 . 
   the differential quadrature method ( dqm ) dates back to bellman et al . 
   @xcite . after the seminal paper of bellman , various test functions have been proposed , among others , spline functions , sinc function , lagrange interpolation polynomials , radial base functions , modified cubic b - splines , see @xcite , etc . 
   shu and richards @xcite have generalized approach of dqm for numerical simulation of incompressible navier - stokes equation . 
   the main goal of this paper is to find numerical solution of initial value system of @xmath3d _ convection - diffusion _ equation with both kinds of boundary conditions ( dirichlet boundary conditions and neumann boundary conditions ) , approximated by dqm with new sets of modified cubic b - splines ( modified extended cubic b - splines , modified exponential cubic b - splines , modified trigonometric cubic b - splines ) as base functions , and so called modified trigonometric cubic - b - spline differential quadrature method ( mtb - dqm ) , modified exponential cubic - b - spline differential quadrature method ( mexp - dqm ) and third modified extended cubic - b - spline differential quadrature method ( mecdq ) . 
   these methods are used to transform the convection diffusion problem into a system of first order odes , in time . 
   the resulting system of odes can be solved by using various time integration algorithm , among them , we prefer ssp - rk54 scheme @xcite due to its reduce storage space , which results in less accumulation errors . 
   the accuracy and adaptability of the method is illustrated by three test problems of two dimensional convection diffusion equations . 
   the rest of the paper is organized into five more sections , which follow this introduction . specifically , section [ sec - metho - decr - tem ] deals with the description of the methods : namely- mtb - dqm , mexp - dqm and mecdq . 
   section [ sec - impli ] is devoted to the procedure for the implementation of describe above these methods for the system together with the boundary conditions as in and . 
   section [ sec - stab ] deals with the stability analysis of the methods . 
   section [ sec - num ] deals with the main goal of the paper is the numerical computation of three test problems . 
   finally , section [ sec - conclu ] concludes the results . 
   the differential quadrature method is an approximation to derivatives of a function is the weighted sum of the functional values at certain nodes @xcite . 
   the weighting coefficients of the derivatives is depend only on grids @xcite . this is the reason for taking the partitions @xmath25 $ ] of the problem domain @xmath26 distributed uniformly as follows : @xmath27 = \ { ( x_i , y_j)\in \omega : h_x = x_{i+1}-x_{i } , h_y = y_{j+1}-y_{j } , i\in \delta_x , j\in \delta_y\},\ ] ] where @xmath28 , and @xmath29 are the discretization steps in both @xmath17 and @xmath18 directions , respectively . 
   that is , a uniform partition in each @xmath30-direction with the following grid points : @xmath31 @xmath32 let @xmath33 be the generic grid point and @xmath34    the @xmath35-th order derivative of @xmath36 , for @xmath37 , with respect to @xmath30 at @xmath33 for @xmath38 is approximated as follows : @xmath39 where the coefficients @xmath40 and @xmath41 , the time dependent unknown quantities , are termed as the weighting functions of the @xmath35th - order derivative , to be computed using various type of base functions .      the trigonometric cubic b - spline function 
   @xmath42 at node @xmath43 in @xmath17 direction , read as @xcite : @xmath44 where @xmath45 , @xmath46 . 
   then the set @xmath47 forms a base over the interval @xmath48 $ ] . setting @xmath49 the values of @xmath50 and its first and second derivatives in the grid point @xmath51 , denoted by @xmath52 , @xmath53 and @xmath54 , respectively , read : @xmath55 the modified trigonometric 
   cubic b - splines base functions are defined as follows @xcite :    @xmath56    now , the set @xmath57 is the base over @xmath48 $ ] . 
   the procedure to define modified trigonometric cubic b - splines in @xmath18 direction , is followed analogously .      in order to evaluate the weighting coefficients @xmath60 of first order partial derivative in eq . 
   , we use the modified trigonometric cubic b - spline @xmath61 , @xmath62 in dq method as base functions . setting @xmath63 and @xmath64 . using mtb - dqm , the approximate values of first order derivative 
   is obtained as follows : @xmath65 setting @xmath66 $ ] , @xmath67 $ ] , and @xmath68 $ ] . eq . reduced to compact matrix form : @xmath69 the coefficient matrix @xmath70 of order @xmath71 can be read from and as : @xmath72\ ] ] and the columns of the matrix @xmath73 read as : @xmath74 = \left [ \begin{array}{c }      2 a_4\\      a_3-a_4\\      0 \\    \vdots   \\            \\    0      \\    0 
   \\ \end{array } \right ] , \im'[2 ] = \left[\begin{array}{c }       a_4 \\      $ 0 $ 
   \\       a_3   \\       $ 0 $ 
   \\     \vdots   \\             \\      $ 0 $ 
   \\ \end{array}\right ] , \ldots , \im'[n_x-1 ] =   \left [ \begin{array}{c }       $ 0 $      \\ 
   \vdots   \\               \\       $ 0 $ 
   \\       a_4   \\       $ 0 $ 
   \\        a_3   \\ \end{array } \right ] , \mbox { and } \im'[n_x ] = \left [ \begin{array}{c }      $ 0 $ \\          \\ 
   \vdots   \\          \\ $ 0 $      \\    2a_4 
   \\   a_3- a_4   \\ \end{array } \right].\ ] ]      the exponential cubic b - splines function @xmath75 at node @xmath43 in @xmath17 direction , reads @xcite : @xmath76 where @xmath77 the set @xmath78 forms a base over @xmath48 $ ] . setting the values of @xmath79 and its first and second derivatives at @xmath51 by @xmath80 , @xmath81 and @xmath82 , respectively . then @xmath83 the modified exponential cubic b - splines base functions 
   are read as :    @xmath84    the set @xmath85 is a base over @xmath48 $ ] . the procedure to define modified trigonometric cubic b - splines in @xmath18 direction , 
   is followed analogously .      setting @xmath86 and @xmath87 for all @xmath88 . 
   using mexp - dqm , the approximate values of the first - order derivative is given by @xmath89 setting @xmath90 $ ] , @xmath67 $ ] , and @xmath91 $ ] , then eq . can be reduced to compact matrix form : @xmath92 let @xmath93 and @xmath94 . using eqns . and , the coefficient matrix @xmath95 of order @xmath71 , read as : @xmath96\ ] ] and the columns of the matrix @xmath97 read : @xmath98 = \left [ \begin{array}{c }      \omega / h_x\\      -\omega / h_x\\      0 \\    \vdots   \\            \\    0      \\    0      \\ \end{array } \right ] , 
   \beth'[2 ] = \left[\begin{array}{c }      \omega/2h_x \\      $ 0 $     \\      -\omega/2h_x   \\ 
   $ 0 $    \\     \vdots   \\             \\      $ 0 $ 
   \\ \end{array}\right ] , \ldots , \beth'[n_x-1 ] =   \left [ \begin{array}{c }       $ 0 $      \\       \vdots   \\               \\       $ 0 $      \\      \omega/2h_x   \\       $ 0 $      \\ 
   \\ \end{array } \right ] , \mbox { and } \beth'[n_x ] = \left [ \begin{array}{c }      $ 0 $ \\ 
   \\ \vdots   \\          \\ $ 0 $      \\   \omega / h_x \\ -\omega / h_x    \\ \end{array } \right].\ ] ]      the extended cubic b - splines function @xmath99 , in the @xmath17 direction and at the knots , reads @xcite : @xmath100 where @xmath101 and @xmath102 , and @xmath103 is a free parameter @xcite . 
   the set @xmath104 forms a base over @xmath48 $ ] . 
   let @xmath105 and @xmath106 , then the values of @xmath107 and its first and second derivatives in the grid point @xmath51 , denoted by @xmath108 , @xmath109 and @xmath110 , respectively , read :    @xmath111    the modified extended cubic b - splines base functions are defined as follows @xcite :    @xmath112    the set @xmath113 is a base over @xmath48 $ ] .      setting @xmath114 and @xmath115 . 
   using mecdq method , the approximate values of the first - order derivative is given by @xmath116 setting @xmath117 $ ] , @xmath67 $ ] , and @xmath118 $ ] . 
   eq . can be re - written in compact matrix form as : @xmath119 using eqns . and 
   , the matrix @xmath120 of order @xmath71 read as : @xmath121\ ] ] and the columns of the matrix @xmath122 read : @xmath123 = \left [ \begin{array}{c }      -1/h_x\\      1/h_x\\      0 \\ 
   \vdots   \\            \\    0 
   \\    0      \\ \end{array } \right ] , \phi'[2 ] = \left[\begin{array}{c }      -1/2h_x \\      $ 0 $     \\      1/2h_x   \\       $ 0 $ 
   \\     \vdots   \\             \\      $ 0 $ 
   \\ \end{array}\right ] , \ldots , \phi'[n_x-1 ] =   \left [ \begin{array}{c }       $ 0 $      \\       \vdots   \\               \\       $ 0 $      \\      -1/2h_x   \\       $ 0 $      \\ 
   1/2h_x    \\ \end{array } \right ] , \mbox { and } \phi'[n_x ] = \left [ \begin{array}{c }      $ 0 $ \\ 
   \\ \vdots   \\          \\ $ 0 $      \\   -1/h_x \\ 1/h_x    \\ 
   \end{array } \right].\ ] ]    using  thomas algorithm " the system , and have been solved for the weighting coefficients @xmath124 , for all @xmath125 .    similarly , the weighting coefficients @xmath126 , in either case , can be computed by employing these modified cubic b - splines in the @xmath18 direction .    using @xmath127 and @xmath126 , the weighting coefficients , @xmath128 and @xmath129 ( for @xmath130 ) 
   can be computed using the shu s recursive formulae @xcite :    @xmath131    where @xmath35 denote @xmath35-th order spatial derivative . in particular 
   , the weighting coefficients @xmath132 of order @xmath3 can be obtained by taking @xmath133 in . 
   after computing the approximate values of first and second order spatial partial derivatives from one of the above three methods , one can re - write eq as follows :    @xmath134    in case of the dirichlet conditions , the solutions on boundaries can directly read from as : @xmath135.\ ] ]    on the other hand , if the boundary conditions are neumann or mixed type , then the solutions at the boundary are obtained by using any above methods ( mtb - dqm , mexp - dqm or mecdq method ) on the boundary , which gives a system of two equations . on solving it we get the desired solution on the boundary as follows :    from eq . with @xmath136 and 
   the neumann boundary conditions at @xmath137 and @xmath138 , we get    @xmath139    in terms of matrix system for @xmath140 , the above equation can be rewritten as @xmath141 \left [ \begin{array}{c } u_{1j } \\ 
   u_{n_x j } \\ 
   \end{array } \right ] = \left [ \begin{array}{c } s_j^a \\ s_j^b \\ \end{array } \right],\ ] ] where @xmath142 and @xmath143 . 
   on solving , for the boundary values @xmath144 and @xmath145 , we get @xmath146    analogously , for the neumann boundary conditions at @xmath147 and @xmath148 , the solutions for the boundary values @xmath149 and @xmath150 can be obtained as : @xmath151 where @xmath152 and @xmath153 .    after implementing the boundary values , eq can be written in compact matrix form as follows : @xmath154 where    1 . 
   @xmath155 $ ] is the solution vector : + @xmath156 , and @xmath157 represents the initial solution vector . 
   2 .   @xmath158 $ ] is the vector of order @xmath159 containing the boundary values , i.e. , + @xmath160 3 . 
   @xmath161 be a square matrix of order @xmath162 given as @xmath163 where @xmath164 and @xmath165 @xmath166 are the block diagonal matrices of the weighting coefficients @xmath58 and @xmath167 , respectively as given below @xmath168 , \mbox { and }     &   ~ b_r = \left [      \begin{array}{cccc }        m_r & o & \ldots & o \\        o & m_r & \ldots   & o \\        \vdots & \vdots & \ddots & \vdots \\        o & o & \ldots   & m_r \\ 
   \end{array }    \right ] , \end{array}\ ] ] where @xmath169 and @xmath170 are the matrices of order @xmath171 , and the sub - matrix @xmath172 of the block diagonal matrix @xmath173 is given by @xmath174 . 
   \end{array}\ ] ]    finally , we adopted ssp - rk54 scheme @xcite to solve the initial value system as : @xmath175 where @xmath176 
   the stability of the method mtb - dqm for @xmath3d convection - diffusion equation depends on the stability of the initial value system of odes as defined in . 
   noticed that whenever the system of odes is unstable , the proposed method for temporal discretization may not converge to the exact solution . moreover , being the exact solution can directly obtained by means of the eigenvalues method , the stability of depends on the eigenvalues of the coefficient matrix @xmath161 @xcite . 
   in fact , the stability region is the set @xmath177 , where @xmath178 is the stability function and @xmath179 is the eigenvalue of the coefficient matrix @xmath161 . the stability region for ssp - rk54 scheme is depicted in ( * ? ? ? 
   * fig.1 ) , from which one can clam that for the stability of the system it is sufficient that @xmath180 for each eigenvalue @xmath179 of the coefficient matrix b. hence , the real part of each eigenvalue is necessarily either zero or negative . 
   it is seen that the eigenvalues of the matrices @xmath181 and @xmath173 @xmath182 have identical nature . 
   therefore , it is sufficient to compute the eigenvalues , @xmath183 and @xmath184 , of the matrices @xmath185 and @xmath186 for different values of grid sizes @xmath187 . the eigenvalues @xmath183 and @xmath184 for @xmath188 has been depicted in figure [ eq - eignv ] . 
   analogously , one can compute the eigenvalues @xmath183 and @xmath184 using mecdq method @xcite or mexp - dqm . 
   it is seen that in either case , @xmath183 and @xmath184 have same nature as in figure [ eq - eignv ] . 
   further , we can get from figure [ eq - eignv ] that each eigenvalue @xmath179 of the matrix @xmath161 as defined in eq . 
   is real and negative . 
   this confirms that the proposed methods produces stable solutions for two dimensional convection - diffusion equations . 
   this section deals with the main goal , the numerical study of three test problems of the initial value system of convection - diffusion equations with both kinds of the boundary conditions has been done by adopting the methods mtb - dqm , mexp - dqm and mecdq method along with the integration ssp - rk54 scheme . 
   the accuracy and the efficiency of the methods have been measured in terms of the discrete error norms : namely- average @xmath189 norm ( @xmath190-error norm ) and the maximum error ( @xmath191 error norm ) . 
   [ ex1 ] consider the initial value system of @xmath3d convection - diffusion equation with @xmath192 , while values of @xmath193 for @xmath194 can be extracted from the exact solution @xmath195 where initial condition is a gaussian pulse with unit hight centered at @xmath196 . 
   we have computed the numerical solution of problem [ ex1 ] for @xmath197 and different values of @xmath198 .    for @xmath199 ^ 2 $ ] : @xmath200 errors and the rate of convergence ( roc ) @xcite at @xmath201 for different values of @xmath202 ( @xmath203 ) 
   has been reported in table [ tab1.1 ] for @xmath204 . 
   table [ tab1.1 ] confirms that the proposed solutions obtained by these methods are more accurate , and approaching towards exact solutions . 
   the behavior of the approximate solution at time @xmath201 taking @xmath205 is depicted in figure [ ex1-fig1.1 ] for @xmath206 and in figure [ ex1-fig1.2 ] for @xmath207 .    for @xmath208 ^ 2 $ ] , time @xmath209 , and grid space step size @xmath210 : the @xmath190 and @xmath200 error norms in the proposed solutions 
   have been compared with the errors in the solutions by various schemes in @xcite in table [ tab1.2 ] for @xmath211 and in table [ tab1.3 ] for @xmath212 . 
   the initial solution and mtb - dqm solutions in @xmath213 ^ 2 $ ] with the parameter values @xmath214 have been depicted in figure [ ex1-fig1.3 ] , and similar behavior is seen from the other two methods . 
   it is evident from the above reports that the proposed results are more accurate as compared to @xcite and approaching towards the exact solutions . 
   l*8l + @xmath71 & & & & & + & @xmath200 @xmath215 & roc @xmath215 & & @xmath200 @xmath215 & roc @xmath215 & & @xmath200 @xmath215 & roc + 5 & 2.91e-03 & & & 2.91e-03 & & & 1.94e-03 & + 10&6.41e-04 & 2.18 & & 6.42e-04&2.181 & & 4.93e-04 & 1.98 + 20&6.50e-05 & 3.30 & & 6.51e-05&3.303 & & 6.05e-05 & 3.03 + 40&7.19e-06 & 3.18 & & 7.18e-06&3.180 & & 6.91e-06 & 3.13 + 80@xmath216 & 7.22e-07 @xmath217 & 3.31 & @xmath218 & 7.00e-07@xmath217&3.358 & @xmath218 & 6.99e-07 @xmath219&3.31 +   + 5 & 4.93e-04 & & & 4.97e-04 & & & 3.20e-04 & + 10&4.86e-05 & 3.34 & & 4.87e-05 & 3.35 & & 4.39e-05&2.87 + 20&4.70e-06 & 3.37 & & 4.70e-06 & 3.37 & & 4.35e-06&3.34 + 40&4.51e-07 & 3.38 & & 4.46e-07 & 3.40 & & 4.50e-07&3.27 + 80@xmath216&4.58e-08@xmath216 & 3.30 @xmath216 & & 3.49e-08 @xmath216&3.67 & & 4.37e-08@xmath216 & 3.36 +    * 3l schemes & @xmath190 & @xmath200 + p - r - adi @xcite & 3.109e-04 @xmath220 @xmath220 & 7.778e-03 + noye @xmath221 tang @xcite & 1.971e-05 & 6.509e-04 + kalila 
   @xcite & 1.597e-05 & 4.447e-04 + kara @xmath221 zhang adi @xcite & 9.218e-06 & 2.500e-04 + tang @xmath221 ge adi@xcite & 9.663e-06 & 2.664e-04 + dehghan @xmath221 mohebbi @xcite@xmath220 @xmath220 & 9.431e-06 & 2.477e-04 + mtb - dqm & 8.026e-12 & 4.327e-08 + mexp - dqm(p=0.0001 ) & 7.030e-12 & 4.154e-08 + mecdq ( @xmath222 ) & 4.504e-12 & 3.343e-08 + mcb - dqm ( @xmath223 ) & 8.269e-12 & 4.388e-08 +    * 3l schemes & @xmath190 & @xmath200 + noye @xmath221 tang @xcite & 1.430e-05 & 4.840e-04 + kalila et al.@xcite & 1.590e-05 & 4.480e-04 + dehghan @xmath221 mohebbi @xcite @xmath220 @xmath220 & 9.480e-06 @xmath220 @xmath220 @xmath220 & 2.469e-04 + mtb - dqm & 8.900e-12 & 4.501e-08 + mexp - dqm(p=0.0001 ) & 9.147e-12 & 4.315e-08 + mecdq ( @xmath224 ) & 4.646e-12 & 3.852e-08 + mcb - dqm ( @xmath223 ) & 9.147e-12 & 4.561e-08 +    [ ex2 ] consider the initial value system of @xmath3d convection - diffusion equation with @xmath225 ^ 2 $ ] with @xmath226 , where @xmath227 and the neumann boundary condition @xmath228,\]]or the dirichlet s conditions can be extracted from the exact solution @xcite : @xmath229    the computation by the proposed methods has been done for different values of @xmath230 and @xmath231 .    for @xmath232 , we take @xmath233 for the solutions at @xmath234 . 
   the rate of convergence and @xmath190 error norms in the proposed solutions has been compared with that of due to fourth - order compact finite difference scheme @xcite for @xmath235 , in table [ tab2.1 ] . 
   it is found that the proposed solutions from either method are more accurate in comparison to @xcite , and are in good agreement with the exact solutions . for @xmath236 
   , we take @xmath237 for the solution at @xmath234 . in table 
   [ tab2.2 ] , the @xmath190 and @xmath200 error norms are compared with that obtained by fourth - order compact finite difference scheme @xcite for @xmath236 , and @xmath238 . 
   rate of convergence have been reported in table [ tab2.3 ] . from tables 
   [ tab2.2 ] and [ tab2.3 ] , we found that the proposed solutions are more accurate as compared to the results in @xcite , while the rate of convergence is linear . 
   the behavior of solutions is depicted in figure [ ex2-fig2.2 ] and [ ex2-fig2.3 ] with @xmath239 for @xmath240 and @xmath241 , respectively . 
   for the same parameter , mentioned above , the numerical solution is obtained by using neumann conditions and reported in table [ tab2.4 ] . 
   the obtained results are in good agreement with the exact solutions , the rate of convergence for @xmath242 is quadratic for each method .    * 2lclcllclcllclcllc + & & & & & & & + @xmath243 & @xmath242 & roc & @xmath190 & roc & & @xmath242 & roc & @xmath190 & roc & & @xmath242 & roc & @xmath190 & roc & & @xmath190 & roc + 0.2 & 4.231e-06 & & 4.969e-10 & & & 1.714e-05 & & 3.584e-09 & & & 4.231e-06 & & 4.999e-10 & & & & + 0.1 & 5.155e-07 & 3.0 & 2.472e-11 & 4.3 & & 5.168e-07 & 5.1 & 2.473e-11 & 7.2 & & 5.153e-07 & 3.0 & 2.478e-11 & 4.3 & & 2.289e-10 & + 0.05 & 5.108e-08 & 3.3 & 9.041e-13 & 4.8 & & 5.105e-08 & 3.3 & 9.044e-13 & 4.8 & & 5.107e-08 & 3.3 & 9.049e-13 & 4.8&&1.621e-11 & 3.82 + 0.025 & 1.147e-08 & 2.2 & 1.922e-14 & 5.6 & & 1.147e-08 & 2.2 & 1.922e-14 & 5.6 & & 1.147e08 & 2.4 & 1.923e-14 & 5.6&&8.652e-13 & 4.23 +   + 0.2 & 7.4031e-06 & & 1.4907e-09 & & & 3.5634e-05 & & 1.5314e-08 & & & 2.0585e-06 & & 6.7290e-11 & & & & + 0.1 & 1.1460e-06 & 2.7 & 1.3103e-10 & 3.5 & & 1.1421e-06 & 5.0 & 1.3148e-10 & 6.9 & & 1.5919e-07 & 3.7 & 1.1624e-12 & 5.9 & & 2.749e-09 & + 0.05 & 1.4578e-07 & 3.0 & 7.5377e-12 & 4.1 & & 1.4582e-07 & 3.0 & 7.5396e-12 & 4.1 & & 1.9012e-08 & 3.1 & 5.4939e-14 & 4.4 & & 2.394e-10 & 3.52 + 0.025 & 1.3902e-08 & 3.4 & 2.7129e-13 & 4.8 & & 1.3904e-08 & 3.4 & 2.7132e-13 & 4.8 & & 6.2220e-09 & 1.6 & 9.0456e-15 & 2.6 & & 1.658e-11 & 3.85 +    * 6l*6l*6l & & & & & & & + @xmath243 & @xmath200 & & @xmath190 & & @xmath200 & & @xmath190 & & @xmath200 & & @xmath190 & & @xmath200 & & @xmath190 + 0.04 @xmath215 & 4.1718e-03@xmath215 & & 1.7833e-03@xmath215 & & 1.7475e-03@xmath215 & & 7.9168e-04 & & 1.2939e-03@xmath215 & & 3.6957e-04 & & 1.1826e-01 @xmath215 & & 4.9331e-03 + 0.02 & 5.2095e-04 & & 6.8410e-05 & & 5.2097e-04 & & 6.8445e-05 & & 6.3735e-04 & & 6.8622e-05 & & 1.5310e-02 & & 4.1351e-04 + 0.01 & 3.2496e-04 & & 2.1508e-05 & & 3.2484e-04 & & 2.1493e-05 & & 2.5394e-04 & & 1.8260e-05 & & 9.4696e-04 & & 2.8405e-05 +    * 5l*5l*5l & + & & & & & + @xmath243 & @xmath242 & roc & @xmath190 & roc & & @xmath242 & roc & @xmath190 & roc & & @xmath242 & roc & @xmath190 & roc + 0.1 & 2.6263e-02 & & 3.6277e-02 & & & 2.6280e-02 & & 3.6670e-02 & & & 2.4464e-02 & & 4.0175e-02 & + 0.05 & 6.8144e-03 & 1.9 & 4.1447e-03 & 3.1 & & 6.8144e-03 & 1.9 & 4.1339e-03 & 3.1 & & 3.3580e-03 & 2.9 & 1.6264e-03 & 4.6 + 0.025 & 1.1634e-03 & 2.6 & 2.7780e-04 & 3.9 & & 1.0719e-03 & 2.7 & 1.9697e-04 & 4.4 & & 8.1776e-04 & 2.0 & 1.0090e-04 & 4.0 + & + & & & & & + 0.01 & 5.0998e-04 & & 3.5191e-03 & & & 5.0738e-04 & & 1.0116e-05 & & & 7.4180e-04 & & 2.5558e-05 & + 0.05 & 1.8776e-04 & 1.4 & 1.8193e-03 & 1.0 & & 1.8776e-04 & 1.4 & 2.9979e-06 & 1.8 & & 2.0972e-04 & 1.8 & 5.1381e-06 & 2.3 + 0.025 & 9.9342e-05 & 0.9 & 9.1372e-04 & 1.0 & & 9.9342e-05 & 0.9 & 7.9462e-07 & 1.9 & & 9.5677e-05 & 1.1 & 8.1188e-07 & 2.7 +    l*4c*5c*5c & + & & & & & + @xmath243 & @xmath242 & roc & @xmath190 & roc & & @xmath242 & roc & @xmath190 & roc & & @xmath242 & roc & @xmath190 & roc + 0.05 & 1.0176e-02 & & 6.7504e-03 & & & 1.0177e-02 & & 6.7513e-03 & & & 9.1549e-03 & & 5.7412e-03 & + 0.025 & 2.2164e-03 & 2.2 & 1.0428e-03 & 2.7 & & 2.2165e-03 & 2.2 & 1.0429e-03 & 2.7 & & 1.9834e-03 & 2.2 & 8.7214e-04 & 2.7 + 0.0125 & 2.9460e-04 & 2.9 & 7.2888e-05 & 3.8 & & 2.9460e-04 & 2.9 & 7.2889e-05 & 3.8 & & 2.5565e-04&3.0 & 5.6518e-05 & 3.9 + & + & & & & & + 0.05 & 1.2070e-01 & & 6.3285e-01 & & & 1.2070e-01 & & 6.3293e-01 & & & 1.0921e-01 & & 6.0454e-01 & + 0.025 & 2.7808e-02 & 2.1 & 6.2302e-02 & 3.3 & & 2.7809e-02 & 2.1 & 6.2304e-02 & 3.3 & & 2.6004e-02 & 2.1 & 5.5004e-02 & 3.5 + 0.0125&6.0348e-03 & 2.2 & 6.4433e-03 & 3.3 & & 6.0348e-03 & 2.2 & 6.4433e-03 & 3.3 & & 5.4733e-03 & 2.2 & 5.5668e-03 & 3.3 + & + & & & & & + 0.05 & 1.4931e-05 & & 5.8044e-08 & & & 1.4932e-05 & & 5.8056e-08 & & & 1.3598e-05 & & 4.8332e-08 & + 0.025 & 3.3009e-06 & 2.2 & 1.0664e-08 & 2.4 & & 3.3009e-06 & 2.2 & 1.0665e-08 & 2.4 & & 2.9824e-06 & 2.2 & 8.7286e-09 & 2.5 + 0.0125 & 4.5947e-07 & 2.8 & 8.0780e-10 & 3.7 & & 4.5947e-07 & 2.8 & 8.0780e-10 & 3.7 & & 3.9840e-07 & 2.9 & 6.0761e-10 & 3.8 + & + & & & & & + 0.05 & 1.7132e-05 & & 6.9905e-08 & & & 1.7132e-05 & & 6.9910e-08 & & & 1.6511e-05 & & 6.5219e-08 & + 0.025 & 3.9931e-06 & 2.1 & 1.4135e-08 & 2.3 & & 3.9932e-06 & 2.1 & 1.4136e-08 & 2.3 & & 3.7892e-06 & 2.1 & 1.2756e-08 & 2.4 + 0.0125 & 8.8931e-07 & 2.2 & 2.7052e-09 & 2.4 & & 8.8931e-07 & 2.2 & 2.7052e-09 & 2.4 & & 8.2511e-07 & 2.2 & 2.3316e-09 & 2.5 +    [ ex3 ] the initial value system of @xmath3d convection - diffusion equation with @xmath244 ^ 2 $ ] and @xmath245 , and @xmath246 where @xmath247 the distribution of the initial solution is depicted in figure [ ex3-fig3.1 ] . 
   the solutions behavior is obtained for the parameter values : @xmath248 and @xmath249 , and is depicted in figure [ ex3-fig3.2 ] due to mtb - dqm , also we noticed the similar characteristics obtained using mexp - dqm and mecdq method . 
   the obtained characteristics agreed well as obtained in @xcite . 
   in this paper , the numerical computations of initial value system of two dimensional _ convection - diffusion _ equations with both kinds of boundary conditions has been done by adopting three methods : modified exponential cubic b - splines dqm , modified trigonometric cubic b - splines dqm , and mecdq method @xcite , which transforms the _ convection - diffusion _ equation into a system of first order ordinary differential equations ( odes ) , in time , which is solved by using ssp - rk54 scheme . 
   the methods are found stable for two space convection - diffusion equation by employing matrix stability analysis method . 
   section [ sec - num ] shows that the proposed solutions are more accurate in comparison to the solutions by various existing schemes , and are in good agreement with the exact solutions . 
   the order of accuracy of the proposed methods for the convection - diffusion problem with dirichlet s boundary conditions is cubic whenever @xmath250 and otherwise it is super linear , in space . on the other hand , 
   the order of accuracy of the proposed methods for the convection - diffusion problem with neumann boundary condition is quadratic with respect to @xmath251 error norms , see table [ tab2.4 ] . 
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   ( left ) of 2d convection - diffusion equation with @xmath238 , @xmath254 for @xmath255 , title="fig:",width=236,height=170 ]   ( left ) of 2d convection - diffusion equation with @xmath238 , @xmath254 for @xmath255 , title="fig:",width=379,height=170 ]         ( left ) of 2d convection - diffusion equation with @xmath238 , @xmath254 for @xmath255 , title="fig:",width=283,height=194 ]   ( left ) of 2d convection - diffusion equation with @xmath238 , @xmath254 for @xmath255 , title="fig:",width=283,height=194 ]    d convection - diffusion equation in example [ ex3 ] at @xmath256 ( left ) and @xmath257 ( right),title="fig:",width=283,height=194 ] d convection - diffusion equation in example [ ex3 ] at @xmath256 ( left ) and @xmath257 ( right),title="fig:",width=283,height=194 ]    d convection - diffusion equation in example [ ex3 ] at @xmath256 ( left ) and @xmath257 ( right),title="fig:",width=283,height=194 ] d convection - diffusion equation in example [ ex3 ] at @xmath256 ( left ) and @xmath257 ( right),title="fig:",width=283,height=194 ]    d convection - diffusion equation in example [ ex3 ] at @xmath256 ( left ) and @xmath257 ( right),title="fig:",width=292,height=194 ] d convection - diffusion equation in example [ ex3 ] at @xmath256 ( left ) and @xmath257 ( right),title="fig:",width=292,height=194 ]
summary: this paper deals with the numerical computations of two space dimensional time dependent _ parabolic partial differential equations _ by adopting adopting an optimal five stage fourth - order strong stability preserving runge - kutta ( ssp - rk54 ) scheme for time discretization , and three methods of differential quadrature with different sets of modified b - splines as base functions , for space discretization : namely- @xmath0 mecdqm : ( dqm with modified extended cubic b - splines ) ; @xmath1 mexp - dqm : dqm with modified exponential cubic b - splines , and @xmath2 mtb - dqm : dqm with modified trigonometric cubic b  splines . 
   specially , we implement these methods on _ convection - diffusion _ equation to convert them into a system of first order ordinary differential equations ( odes ) , in time . 
   the resulting system of odes can be solved using any time integration method , while we prefer ssp - rk54 scheme . 
   all the three methods are found stable for two space convection - diffusion equation by employing matrix stability analysis method . 
   the accuracy and validity of the methods are confirmed by three test problems of two dimensional _ convection - diffusion _ equation , which shows that the proposed approximate solutions by any of the method are in good agreement with the exact solutions . 
   convection - diffusion equation , modified trigonometric cubic - b - splines , modified exponential cubic - b - splines , modified extended cubic - b - splines , differential quadrature method , ssp - rk54 scheme , thomas algorithm