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flow and transport are fundamental in many geo - engineering applications , including oil and gas recovery from hydrocarbon reservoirs , groundwater contamination and sustainable use of groundwater resources , storing greenhouse gases ( e.g. co@xmath1 ) or radioactive waste in the subsurface , or mining heat from geothermal reservoirs . in porous media , a single phase transport process is described by the equation @xmath2,\end{aligned}\ ] ] where @xmath3 is an open domain of @xmath4 , @xmath5 is the porosity ( void fraction ) of the rock , and @xmath6 is the symmetric dispersion tensor , @xmath7 is the unknown concentration of the contaminant , @xmath8 the darcy s velocity and @xmath9 the reaction and source term . + finite element , finite volume or the combination finite element - finite volume methods are mostly used for space discretization of the problem ( [ adr ] ) while explicit euler , semi implicit and fully implicit methods are usually used for time discretization @xcite . due to time step size constraints , fully implicit scheme is more popular for time discretization for quite a long time compared to explicit euler and semi implicit methods . this method , however , need at each time step a solution of large systems of nonlinear equations . this can be the bottleneck in computations . in recent years , exponential integrators have become an attractive alternative in many evolutions equations @xcite . in contrast to classical methods , they do not require the solution of large linear systems . instead they make explicit use of the matrix exponential and related matrix functions . the price to pay is the computing of the matrix exponential functions of the non diagonal matrices , which has revived interest and significance progresses nowadays @xcite .    in this work , we combine a finite volume method with the first order exponential time differencing scheme of order 1 ( etd1 ) . although both discretization techniques have been together used for solving evolutionary problems like ( [ adr ] ) ( see @xcite ) , a proper combination of rigorous proof of them has been lacking so far . the paper is organised as follows . in section  [ sec : semi ] we present the semi group formulation . in section  [ sec : fvm ] and section  [ sec : etd ] we present the space and time discretization . we then state and discuss our main result in section  [ sec : th1 ] and present the proof of the convergence theorem . we end by presenting some simulations in section  [ sec : simulation ] , these are applied both to a linear example where we can compute an exact solution as well as a more realistic model ( spe10 case @xcite ) . let us start by presenting briefly the notation of the main function spaces and norms that we use in this paper . we denote by @xmath10 the norm associated to the inner product @xmath11 of the hilbert space @xmath12 . for a banach space @xmath13 we denote by @xmath14 the norm of @xmath13 and @xmath15 the set of bounded linear mapping from @xmath13 to @xmath13 to @xmath16 . for the sake of simplicity , without loss of generality we use the homogeneous dirichlet boundary condition and assume that the porosity @xmath5 is constant . during our simulation , we will also investigate the general boundary conditions and non constant porosity @xmath5 . we assume that the darcy velocity @xmath8 is known , and satisfies the following mass conservation for incompressible fluids @xmath17 the model problem ( [ adr ] ) is reformulated as : find the concentration @xmath18 such that @xmath19   \\   x(\mathbf{x},0)=x_{0 } \quad \quad \quad \quad & \mathbf{x }   \in \omega\\ x(\mathbf{x},t ) = 0    \quad \quad \quad \quad & ( \mathbf{x},t ) \in \partial \omega \times \left[0,t \right ]   \end{cases }   \end{aligned}\ ] ] where @xmath20 for well posedness of ( [ adr6 ] ) , we assume that @xmath6 is symmetric , @xmath21 and there exists a positive constant @xmath22 such that @xmath23 set @xmath24 , the bilinear form associated to the operator @xmath25 is given by @xmath26 according to grding s inequality ( see @xcite ) , there exists two positive constants @xmath27 and @xmath28 such that @xmath29 by adding @xmath30 in both side of the first equation of ( [ adr ] ) we have a new operator that we still call @xmath31 corresponding to the new bilinear form that we still call @xmath32 such that the following coercivity property holds @xmath33 we will still call the right hand side of the first equation of ( [ adr6 ] ) @xmath9 . by green s formula we have @xmath34 where @xmath35 the domain of the operator @xmath31 . therefore the weak form of ( [ adr6 ] ) is to find the function @xmath36 such that @xmath37   \\ x(t)=x_{0}. \end{cases}\end{aligned}\ ] ] the @xmath38ellipticity ( [ ellip ] ) implies that @xmath39 is a sectorial on @xmath40 ( see @xcite ) i.e. there exists @xmath41 such that @xmath42 where @xmath43 . then @xmath39 is the infinitesimal generator of bounded analytic semigroups @xmath44 on @xmath40 such that @xmath45 where @xmath46 denotes a path that surrounds the spectrum of @xmath47 . the coercivity property in ( [ ellip ] ) implies also that the set of the real part of the spectrum of @xmath31 is non negative , which allows the definition of the fractional power of @xmath31 as : for any @xmath48 @xmath49 where @xmath50 is the gamma function of @xmath51 @xcite . we denote by @xmath52 the norm of the space @xmath53 .    for the nonlinear reaction term we make the following assumption    * [ lipschitz condition for @xmath9 ] * + [ alip ] the nonlinear @xmath9 is continuous and differentiable respect to @xmath7 with respect to the variable @xmath7 , with bounded differential . therefore there exists a positive constant @xmath55 such that @xmath56 \\ \newline\\   \vert r(y)-r(z ) \vert \leq l \vert y - z\vert\;\;\;\forall y , z \in l^{2}(\omega ) , \end{array}\right.\end{aligned}\ ] ]    where with a slight abuse of notation , @xmath9 denotes the nonlinear operator @xmath57 .    by duhamel s principle we may represent solutions of ( [ adr6 ] ) by the following integral equation @xmath58 . \ ] ] applying the contraction mapping principle in the topology of the banach space @xmath59,h^{1}(\omega))$ ] to the integral equation ( [ mild ] ) ( * ? ? ? * theorem 3.3.3 ) or ( * ? ? ? * theorem 6.3.1 ) ensure the existence and uniqueness of @xmath7 . a cell  centred finite volume methods for heterogeneous and anisotropic diffusion problems remains a challenging problem . an active area of research is to make the approximation of the diffusion flux more efficient and simple as possible ( see @xcite for the references ) . the finite volume method is widely applied when the differential equations are in divergence form . to obtain a finite volume discretization , the domain @xmath3 is subdivided into subdomains @xmath60 being the corresponding set of indices , called control volumes or control domains such that the collection of all those subdomains forms a partition of @xmath3 . the common feature of all finite volume methods is to integrate the equation over each control volume @xmath61 and apply gauss s divergence theorem to convert the volume integral to a surface integral . for our parabolic problem ( [ adr6 ] ) , finite volume methods differ in the way they approximate the diffusion flux @xmath62 . two techniques are mostly used : the finite volume with two - point flux approximation ( tpfa ) ( see @xcite ) and the finite volume with multi - point flux approximations ( mpfa)(@xcite ) . an advantage of the two - point approximation is that it provides monotonicity properties , under the form of a local maximum principle . it is efficient and mostly used in industrial simulations . in this paper we use the tpfa as developed in ( * ? ? ? * page 815 ) . the main drawback of tpfa is that it is applicable in the so called `` admissible mesh '' and not in general mesh . [ admissiblemesh ] [ * an admissible mesh @xcite * ]    an admissible mesh @xmath63 for problem ( [ adr6 ] ) with the full diffusion tensor @xmath6 is given by :    * a set @xmath64 of control volumes such that @xmath65 with the corresponding local inner product induced by @xmath66 where @xmath67 * the corresponding set of center points @xmath68 such that 1 . @xmath70 is the intersection of the straight lines perpendicular to the boundary of @xmath71 with respect to the inner product induced by @xmath66 . let @xmath72 be the maximum mesh size of @xmath63 . we denote by @xmath73 a dual delaunay triangulation of @xmath63 i.e. the delaunay triangulation where @xmath74 is the set of vertices . let us illustre definition  [ admissiblemesh ] to make it more understandable . * in the case where the diffusion tensor @xmath6 is diagonal and @xmath3 is a rectangular or parallelepiped domain , any rectangular grid ( @xmath75 ) or parallelepiped grid ( @xmath76 ) is an admissible mesh . the set @xmath77 is the set of centers of gravity of the rectangular grid or parallelepiped grid . the inner product induced locally by @xmath66 is equivalent to the standard inner product corresponding to the euclidean norm @xmath78 . this mesh will yield a 5-point scheme ( @xmath75 ) and 7-point scheme ( @xmath76 ) for our model problem ( [ adr6 ] ) . * if @xmath75 , for isotropic and heterogeneous media ( @xmath79 , @xmath80 being the identity matrix of dimension @xmath81 ) we can define a triangular admissible mesh @xmath82 to be a family of open triangular disjoint subsets of @xmath3 such that two triangles having a common edge have also two common vertices . the angles of the triangles are assumed to be less than @xmath83 to allow the orthogonal bisectors to intersect inside each triangle , thus naturally defining the center point @xmath70 of the control volume @xmath71 . the finite volume scheme defined on such mesh will yield a 4-point scheme for our model problem ( [ adr6 ] ) . the inner product induced locally by @xmath66 is equivalence to the standard inner product corresponding to the euclidean norm @xmath78 .    to make notation easier , we will identify @xmath63 to @xmath84 , then to say @xmath85 we will say @xmath86 . consider the modified model problem of ( [ adr6 ] ) where @xmath30 is added on both sides of the first equation of problem ( [ adr6 ] ) , @xmath27 is defined in ( [ coer ] ) . consider an admissible mesh @xmath63 in the sense of definition  [ admissiblemesh ] . denote by @xmath87 the set of edges of control volume of @xmath88 the set of interior edges of control volume of @xmath89 , @xmath90 the approximation of @xmath7 at time @xmath91 at the center ( or at any point ) of the control volume @xmath92 and @xmath93 the approximation of @xmath7 at time @xmath91 at the center ( or at any point)of the edge @xmath94 . for a control volume @xmath92 , denote by @xmath95 the set of edges of @xmath96 , @xmath97 the lebesgue measure of the control volume @xmath92 .    as in @xcite , integration over any control volume @xmath92 , using the divergence theorem to convert the integral over @xmath96 to a surface integral , finite differences for the diffusion flux approximation @xcite and the upwind technique for the advection flux approximation yields @xmath98 here @xmath99 is the normal unit vector to @xmath100 outward to @xmath96 , @xmath101 is the lebesgue measure of the edge @xmath102 and @xmath103 the distance between the center of @xmath96 and the edge @xmath104 . since the flux is continuous at the interface of two control volumes @xmath96 and @xmath105 ( denoted by @xmath106 ) we therefore have @xmath107 for @xmath108 , which yields @xmath109 with @xmath110 where @xmath111 is the distance between the center of @xmath96 and center of @xmath105 . we will also denote by @xmath112 the distance @xmath111 or @xmath103 for @xmath113 or @xmath114 respectively . notice that for @xmath115 , we can also write @xmath116 with @xmath117 the upwind term for advection flux @xmath118 is defined as @xmath119 we can write @xmath118 as @xmath120 where @xmath121 .    the finite volume space discretization for the model problem ( [ adr6 ] ) is given by @xmath122 the scheme ( [ fvfd ] ) clearly indicates the affinity of the finite volume method to the finite difference method . however , for the subsequent analysis it is more convenient to rewrite scheme ( [ fvfd ] ) in a discrete variational form . multiplying the first equation of ( [ fvfd ] ) by arbitrary numbers @xmath123 and summing the results over all control volume in @xmath63 yields @xmath124 v_{i}=   \underset{i \in \mathcal{t}}{\sum } \mathrm{mes}(i)\,r(x_{i}(t))v_{i},\\ \newline\\ r(x_{i}(t)):=r(\mathbf{x}_{i},t , x_{i}(t ) ) . \end{array}\right.\end{aligned}\ ] ] let @xmath125 denote the space of continuous functions that are piecewise linear over the delaunay triangulation @xmath73 ( dual of @xmath63 ) , then the values @xmath90 and @xmath126 can be interpolated in @xmath127 . there are unique functions @xmath128 such that @xmath129 and @xmath130 for all @xmath131 , where @xmath70 is a center of the control volume @xmath131 ( @xmath70 is also a vertex in @xmath73 ) . denote by @xmath132 the bilinear form defined by @xmath133 and by @xmath134 the scalar product on @xmath135 defined by @xmath136 the corresponding norm is the discrete @xmath137 norm denoted by @xmath138 . it is proved in @xcite that @xmath138 is equivalent to the @xmath137 norm @xmath139 in @xmath127 when the mesh @xmath63 is regular . + previous results allow us to write the following variational form of our finite volume scheme ( [ fvfd1 ] ) . @xmath140,\\ \newline\\ x_{h}(0)=x_{h\,0},\\ \end{array}\right.\end{aligned}\ ] ] with @xmath141 consider the operator @xmath142 such that @xmath143 the semidiscrete solution in @xmath127 is then given by : find @xmath144 such that @xmath145 \\ \newline\\ x_{h}(0)=x_{0 h } \end{array}\right.\end{aligned}\ ] ] where @xmath146 is the orthogonal projection defined from @xmath147 to @xmath127 by @xmath148 in order to give the corresponding mild form of ( [ adrh ] ) and build the exponential integrator scheme let us define the discrete @xmath149 norm . * [ discrete @xmath149 norm @xcite ] * + [ seminorm ] let @xmath63 be an admissible finite volume mesh in the sense of definition  [ admissiblemesh ] . let @xmath150 be the space of the functions constant in each control volume of @xmath63 . for @xmath151 , the discrete @xmath149 norm of @xmath152 is defined by @xmath153 where @xmath154    during our analysis we make the following assumption as in ( * ? ? ? * page 815 ) . [ meshassumption ] [ * regularity of @xmath6 , @xmath8 and the admissible mesh @xmath63 * ] + we assume that @xmath6 is bounded , @xmath155 and that there exists @xmath156 and @xmath157 such that @xmath158    assumption  [ meshassumption ] allows the following @xmath159 ellipticity of @xmath132 . [ coert ] under the regularity of the admissible mesh @xmath63 in assumption  [ meshassumption ] , there exists a constant @xmath48 such that @xmath160    let @xmath161 and @xmath162 the bilinear forms defined in @xmath163 by @xmath164 using assumption  [ meshassumption ] , mainly the regularity of @xmath63 and the fact that the coefficients of the diffusion tensor @xmath6 are bounded , there exists two constants @xmath165 and @xmath166 such that @xmath167 and @xmath168 so that @xmath169 where @xmath170 is defined in ( [ transn11 ] ) and ( [ transn1 ] ) . using the fact that the transmissibility given in ( [ interf ] ) is symmetric , i.e. @xmath171 and reorganizing the summation , we therefore have @xmath172 let use some important results from @xcite . indeed as in @xcite reordering the summation over the set of edges yields @xmath173 where @xmath174 since @xmath175 we therefore have @xmath176 since we have assumed divergence - free flow . then @xmath177 we also have @xmath178 therefore @xmath179    the following @xmath159 ellipticity of @xmath132 implies that @xmath180 is a sectorial on @xmath40 ( uniformly in @xmath72 ) i.e. there exists @xmath41 , such that @xmath181 where @xmath43 . the discrete operator @xmath180 therefore is the infinitesimal generator of bounded analytic semigroup ( or exponential operator ) @xmath182 on @xmath127 such that @xmath183 where @xmath184 denotes a path that surrounds the spectrum of @xmath180 . as for the continuous case , duhamel s principle implies that the solution of ( [ adrh ] ) is represented by the following integral equations ( mild form ) @xmath185.\end{aligned}\ ] ] for simplicity we consider a constant time - step @xmath186 . at time @xmath187 , $ ] the mild solution ( [ adrd ] ) is given by @xmath188 then , given the solution @xmath189 at the time @xmath190 , we can construct the corresponding solution at @xmath191 as @xmath192 note that the expression in ( [ exactnu ] ) is still an exact form of @xmath189 . the idea behind exponential time differencing is to approximate @xmath193 by a suitable polynomial  @xcite . we consider the simplest case where @xmath193 is approximated by the constant @xmath194 and the corresponding scheme ( etd1 ) is given by @xmath195 where @xmath196 note that the etd1 scheme in ( [ etd ] ) can be rewritten as @xmath197 this new expression has the advantage that it is computationally more efficient as only one matrix exponential function needs to be evaluated at each step . we assume that the unique mild solution @xmath7 of adr problem ( [ adr6 ] ) is the classical solution of ( [ adr6 ] ) i.e. @xmath7 is twice continuously differentiable with respect to @xmath198 and differentiable with respect to @xmath91 . this assumption is necessary to achieve optimal orders of convergence in time and space . [ t1 ] consider the mild solution @xmath7 of the adr ( [ adr6 ] ) and the numerical solution ( [ etd1 ] ) given by the etd1 scheme . assume that assumption  [ meshassumption ] is satisfied and the reaction function @xmath9 satisfies assumption  [ alip ] . set @xmath199 , assume that @xmath200 , then the following estimate holds : @xmath201 where @xmath202    the proof follows in section  [ proofd ] , but we need some preparatory results first . [ prop111 ] @xmath203 $ ]    let @xmath89 be an admissible mesh in the sense of definition  [ admissiblemesh ] and @xmath73 its dual delaunay triangulation ( @xmath204 are vertices of @xmath73 ) . let @xmath205 defined by @xmath206 where @xmath207 is the nodal basis corresponding to @xmath208 in the sense of finite element method @xmath209 . if @xmath210 , then there exists a positive constant @xmath211 such that the following estimate holds @xmath212 if @xmath213,c^{2}(\overline{\omega}))$ ] , then @xmath214.\end{aligned}\ ] ]    see ( * ? ? ? * section 3.4 , theorem 3.29 , page 139 or exercise 3.25 page 147 ) or ( * ? ? ? * theorem 17.1 , page 132 ) .    * [ smoothing properties of the semi group@xcite ] * + [ prop1 ] let @xmath215 and @xmath216 , then there exists @xmath217 such that @xmath218 in addition , the following results hold @xmath219 @xmath220 @xmath221 where @xmath222 , @xmath223 . [ lemmad1 ] let @xmath7 be the mild solution of ( [ adr6 ] ) given in ( [ mild ] ) . let @xmath224,\;t_{1}<t_{2}$ ] . the following estimates hold :    * \(i ) if @xmath200 then @xmath225 for @xmath226 small enough . * \(ii ) if @xmath200 and @xmath9 satisfies the lipschitz condition in ( [ lip ] ) then @xmath227    * part ( i)*.    consider the difference @xmath228 so that @xmath229 we estimate each of the terms @xmath230 and @xmath231 . for @xmath230 , using proposition [ prop1 ] yields @xmath232    for the term @xmath233 , we have @xmath234 with @xmath235    we now estimate each term @xmath236 and @xmath237 . for @xmath238 @xmath239 for @xmath226 small enough , using proposition [ prop1 ] yields @xmath240    for @xmath241 , using the fact that the semigroup is bounded , we have @xmath242 hence @xmath243 combining previous estimations of @xmath230 and @xmath244 ends the proof of part ( i ) . * proof of part ( ii ) * we consider again the difference in ( [ ddif ] ) . the difference with the proof of part ( i ) comes from the estimation of @xmath245 . this time we write @xmath246 if @xmath9 satisfies the lipschitz condition given in ( [ lip ] ) , then using the result in part ( i ) together with proposition  [ prop1 ] yields @xmath247 for @xmath226 small enough . we also have @xmath248 using the two transformations @xmath249 we find @xmath250 the estimate of @xmath245 ends the proof . [ lemmad3 ] [ @xcite ] + let the sequence @xmath251 . if the sequence of nonnegative numbers @xmath252 satisfies the inequality @xmath253 for @xmath254 and @xmath255 , then the following estimate holds : @xmath256 where the constant @xmath257 depends on @xmath258 .      in the followng proof , @xmath259 and @xmath260 are positive constants . we use the equivalence of the norms @xmath261 and @xmath138 in @xmath127 as the mesh @xmath63 is regular @xcite .    using the triangle inequality yields @xmath262 let us estimate @xmath263 . integrating the shifted version ( by adding @xmath30 in both size ) of equation ( [ adr6 ] ) over each control volume @xmath264 and using the divergence theorem yields @xmath265 for @xmath266,\;\;i \in \mathcal{t}$ ] and @xmath102 using the same notation as in @xcite , let us set @xmath267 , \\ \newline\\ r_{i,\sigma}(t)= \dfrac{1}{\mathrm{mes}(\sigma)}\left [ q_{i,\sigma}x_{i,+}-\int_{\sigma } \mathbf{q } x(t)\cdot   \mathbf{n}_{\sigma}\right ] , \\ \newline\\ \rho_{i}(t)=x(\mathbf{x}_{i},t)-\dfrac{1}{\mathrm{mes}(i)}\int_{i}x(\mathbf{x},t)d\mathbf{x},\\ \newline\\ \varrho_{i}(t)= \dfrac{1}{\mathrm{mes}(i)}\int_{i } r(\mathbf{x},t , x(\mathbf{x},t))d\mathbf{x}-r(\mathbf{x}_{i},t , x_{i}(t ) ) . \end{array}\right.\end{aligned}\ ] ] assuming that the unique solution @xmath7 of ( [ adr6 ] ) is the regular , taylor expansion yields @xmath268 using assumption  [ meshassumption ] , mainly the fact that the coefficients of the diffusion tensor @xmath6 and the darcy s velocity @xmath8 are regular , we can therefore used the following similar results in @xcite for general elliptic operators @xmath269 using the fact that @xmath9 is differentiable with respect to @xmath7 and @xmath198 , taylor expansion respectively yields @xmath270 where @xmath271 if the solution @xmath7 is differentiable with respect @xmath198 , one more taylor expansion yields @xmath272 using the lipschitz condition ( [ lip ] ) yields @xmath273 subtracting the first equation of ( [ fvfd ] ) from ( [ vdadr ] ) and using previous expressions yields @xmath274 with @xmath275 , \\ \newline\\ g_{i,\sigma}(t)=-\tau_{\sigma}(e_{j}(t)-e_{i}(t)),\quad \quad \sigma= i \vert j , \\ \newline\\ g_{i,\sigma}(t)=\tau_{\sigma}e_{i}(t),\quad   \sigma \in\mathcal{e}_{i}\cap \partial \omega,\\ \newline\\ w_{i,\sigma}(t)=q_{i,\sigma}(x(\mathbf{x}_{\sigma,+},t)-x_{\sigma,+}(t ) ) , \end{array}\right.\end{aligned}\ ] ] and @xmath276 multipling equation ( [ derr1 ] ) by @xmath277 and summing for @xmath131 yields @xmath278 \\ \newline\\    = \underset{i \in \mathcal{t}}{\sum}e_{i}(t)\left [ \int_{i}\left(r(\mathbf{x},t , x(\mathbf{x},t))-r(\mathbf{x}_{i},t , x_{i}(t))\right)d\mathbf{x } \right ] \\ \newline\\    + \underset{i \in \mathcal{t}}{\sum}\left[c_{0 } \,\mathrm{mes}(i)\rho_{i}(t)e_{i}(t)-\underset{\sigma \in \mathcal{e}_{i}}{\sum}\mathrm{mes}(\sigma)e_{i}(t)(r_{i,\sigma}(t)+r_{i,\sigma}(t))-e_{i}(t)s_{i}(t)\right ] . \end{array}\right.\end{aligned}\ ] ] let @xmath279 a piecewise constant function defined by @xmath280.\end{aligned}\ ] ]    as in the proof of theorem [ coert ] , using the fact that @xmath169 where @xmath170 is defined in ( [ transn11 ] ) and ( [ transn1 ] ) yields    @xmath281    where @xmath282 setting @xmath283 as in the proof of theorem [ coert ] , important results from @xcite yields @xmath284 using ( [ flux2 ] ) and ( [ react ] ) in the expression ( [ derr2 ] ) yields @xmath285 the continuity of the diffusion and advection flux at each interface yields @xmath286 set @xmath287 using the relation ( [ conti ] ) , the cauchy - schwarz inequality as in @xcite for stationary elliptic problems and reordering the summation over the edges yields @xmath288 using the fact that @xmath289 and relation ( [ flux1 ] ) yields @xmath290 for any constant @xmath217 , young s inequality yields @xmath291 using expression ( [ derr4 ] ) and ( [ yp ] ) in expression ( [ derr3 ] ) yields @xmath292 \leqslant ( c_{4}'/2 + 1)\vert e_{\mathcal{t}}(t)\vert_{0,h}^{2}+ c_{6}\,h^{2}\\ \newline\\ c_{6}=c_{6}(c_{1 } , c_{3},c_{3}',c_{4},c_{5 } ) . \end{array}\right.\end{aligned}\ ] ] bounding the left hand side of expression ( [ derr5 ] ) below yields @xmath293.\end{aligned}\ ] ] integrating both side of expression ( [ derr6 ] ) through interval @xmath294,\ ; 0\leq t\leq t $ ] yields @xmath295.\end{aligned}\ ] ] applying the discrete gronwall lemma  [ lemmad3 ] yields @xmath296 then @xmath297 if @xmath199 , we have @xmath298 ( see @xcite ) , we therefore have @xmath299 let us estimate @xmath233 . from ( [ mmild ] ) and ( [ etd ] ) we have @xmath300 and @xmath301 the smoothing properties of the semigroup @xmath302 in proposition  [ prop1 ] and the equivalence @xmath303 in @xmath127 yields @xmath304 and @xmath305 for @xmath306 $ ] , using proposition  [ prop111 ] yields @xmath307 since @xmath308 , the equivalence @xmath309 and the uniform estimate of the term @xmath263 in @xmath310 $ ] yields @xmath311 using the lipschitz condition ( [ lip ] ) with ( [ errori ] ) and ( [ errorii ] ) yields @xmath312 we also have @xmath313 using lemma [ lemmad1 ] and the lipschitz condition ( [ lip ] ) yields @xmath314 using lipschitz condition ( [ lip ] ) , proposition  [ prop111 ] and the equivalence @xmath315 in @xmath127 @xmath316 then @xmath317 combining estimates @xmath263 and @xmath233 yields @xmath318 applying the discrete gronwall lemma  [ lemmad3 ] in ( [ lastest ] ) ends the proof . the key point in ( etd1 ) scheme is the computing of the exponential function , the so called @xmath319 . it is well known that a standard pad approximation for a matrix exponential is not an efficient method for large scale problems  @xcite . here we use the real fast leja points and the krylov subspace techniques to evaluate the action of the exponential matrix function @xmath320 on a vector @xmath321 , instead of computing the full exponential function @xmath322 as in a standard padapproximation . the details of the krylov subspace technique are given in @xcite while more information about the real fast leja points can be found in @xcite . the aims here is to analyse the convergence in space by comparing the numerical solution to the exact solution for a linear problem . we consider here a linear reaction diffusion with exact solution given in @xcite . the domain is defined as @xmath323 , @xmath324 , @xmath325 . the initial time is given as @xmath326 . this is necessary because the exact solution is not defined at the origin and at @xmath327 . the dispersion tensor @xmath6 is diagonal and heterogeneous with coefficients given by @xmath328 a curved the velocity field ( figure  [ fig01a ] ) is given explicitly by @xmath329 where @xmath330 and @xmath331 . the finite volume mesh @xmath63 is the rectangular grid . initial and boundary conditions are taken according to the exact solution  @xcite , assuming an instantaneous release at a point @xmath332 . we take a fixed time - step of @xmath333 . we use here the real the fast lja to compute the action of the exponential function @xmath319 . the final time here is @xmath334 . the domain is @xmath336\times [ 0,l_{2}]\times [ 0 , l_{3}]$ ] , the finite volume mesh @xmath63 is the set of parallelepipeds with space discretization stepsize @xmath337 . dimensions are @xmath338,@xmath339 . we use for the absolute tolerance error @xmath340 and @xmath341 for the krylov subspace dimension .    to have the velocity field we solve with our finite volume scheme the system @xmath342 where the dynamical viscosity is @xmath343 and the diagonal permeability tensor @xmath344 comes from @xcite . we use the dirichlet boundary at @xmath345\right\rbrace \cup\left\lbrace \left\lbrace l_{1 } \right\rbrace \times\left\lbrace l_{2}\right \rbrace\times\left[0,l_{3}\right]\right \rbrace,\ ] ] and homogenous neumann boundary elsewhere such that @xmath346\\ \newline\\   7997.92\;\text { psi}\quad\text{in}\quad\left\lbrace l_{1 } \right\rbrace \times\left\lbrace l_{2 } \right\rbrace\times \left[0,l_{3}\right ]   \end{array}\right . \newline\\ - k\,\nabla p ( \mathbf{x},t)\,\cdot\mathbf{n } & = & 0\quad\text{in}\quad\gamma_{n}= \partial \omega \backslash\gamma_{d}.\end{aligned}\ ] ] for the nonlinear advection diffusion reaction we use the following boundary conditions @xmath347\right\rbrace\times \left[0,t \right ]   \newline\\ x&= & 1\quad\text{in}\quad \left\lbrace\left\lbrace l_{1 } \right\rbrace \times \left\lbrace l_{2 } \right\rbrace\times\left[0,l_{3}\right]\right\rbrace\times \left[0,t \right]\\ \newline\\ -(\mathbf{d}\nabla x)(\mathbf{x},t)\cdot\mathbf{n } & = & 0\quad\text{in}\quad\gamma_{n } \times \left[0,t \right ] \newline\\ x_{0}&=&0\quad \text{in}\quad\omega \quad(\text{initial solution})\end{aligned}\ ] ] where @xmath348 is the unit outward normal vector to @xmath349 . the velocity field obtained in ( [ vel ] ) is piecewise constant in the mesh @xmath63 . for reaction function , we use the classical langmuir sorption isotherm given by @xmath350 , with @xmath351 . this reaction function is obviously lipschitz . the reference or exact solution is the numerical solution with the smaller time step size ( @xmath352 ) . the final time here is @xmath353 seconds . we take a uniform dispersion(diffusion ) tensor of @xmath354 . figure  [ fig01b ] shows a convergence of order @xmath355 for the time discretisation , which agrees with the predicted order in theorem  [ t1 ] even we do nt have any information about the regularity of the exact solution . in fact to obtain the optimal order of convergence in time , the exact solution should be differentiable respect to the time , which may be the case here since the initial solution is infinitely differentiable and we are dealing with parabolic problem . m.  a. christie and m.  j. blunt , tenth spe comparative solution project : a comparison of upscaling techniques , 4(4 ) ( 2001 ) 308317 . a.   tambue , g.  j.lord , and s.   geiger . an exponential integrator for advection - 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we analyze the finite volume spatial convergence combined with exponential time differencing of order one ( etd1 ) for temporal discretization of a non linear advection  diffusion  reaction equation ( adr ) modeling transport in porous media . 
 we derive the @xmath0 estimate under the assumption that the non linear reaction is globally lipschitz . 
 we illustrate the theoretical proof by some applications in 2d and 3d highly anisotropic and heterogeneous porous media including the spe10 case @xcite . 
 we compute the exponential of the non diagonal matrices arriving in the finite volume spatial discretization with the fast lja points technique and the krylov subspace technique . 
 finite volume , exponential integrator , porous media , advection  diffusion  reaction