Semi-analytical solutions of contaminant transport equation with nonlinear sorption in 1D

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1 Semi-analytical solutions of contaminant transport equation with nonlinear sorption in D Peter Frolkovič and Jozef Kačur November 9, 24 Abstract A new method to determine semi-analytical solutions of one-dimensional contaminant transport problem with nonlinear sorption is described. This method is based on operator splitting approach where the convective transport is solved exactly and the diffusive transport by finite volume method. The exact solutions for all sorption isotherms of Freundlich and Langmuir type are presented for the case of piecewise constant initial profile and zero diffusion. Very precise numerical results for transport with small diffusion can be obtained even for larger time steps (e.g., when the CFL condition failed). Keywords: contaminant transport, nonlinear sorption, Freundlich isotherm, finite volume method AMS Subject Classification 35K65, 35L67, 65M25 Mathematical model The main goal of this paper is to construct precise semi-analytical solutions of nonlinear convection-diffusion problem with adsorption t F (u) + v(x) x u x (D(x) x u) = () for x (, L), t >, with initial condition and boundary conditions u(x, ) = U (x) (2) u(, t) = C (t), x u(l, t) =. (3) This work was funded by the Federal Ministry of Economics and Technology (BMWi) under the contract number 2 E and the second author was also partially supported by grants MSM 262 and GARC 2/3/57.

2 Here, D(x) D >, v(x) v >, F () =, F (s) δ, where one can assume that δ =. The mathematical model () - (3) can represent contaminant transport in equilibrium mode with the sorption isotherm Ψ(u) = F (u) u, see, e.g., [9]. The corresponding mathematical model is t u + v(x) x u x (D(x) x u) + t S =, t S = k(ψ(u) S), where S = Ψ(u) for k. The most common forms of (nonlinear) sorption isotherms are Freundlich and Langmuir Ψ(u) = au p, a, p > (4) Ψ(u) = but one can consider also isotherms of mixed type au, a, b >, (5) + bu Ψ(u) = The equation () can be written in the form, aup + bu p. (6) F (u) t u + v(x) x u x (D(x) x u) =, (7) where the function F (u) can be viewed as a (nonlinear) retardation factor of the convective and diffusive transport. In the case of F () = (i.e., for (4) and (6) with < p < ), the solution of () can have a sharp front with finite speed of propagation [27]. Moreover, the convective term can be dominant and thus the creation of shocks can be expected even for smooth initial data. Nevertheless, due to the presence of (small) diffusion D >, the solution of () - (3) is regular and sharp shocks (as known for hyperbolic problems) can not develop in a finite time - see, e.g., [5]. Precise numerical solutions of () are required, for instance, if one wishes to solve inverse problems (e.g., to determine the diffusion D and the sorption isotherm) or to test numerical methods for this type of problems. The most numerical methods so far are based on regularisation and/or upwinding procedures and they can produce undesirable artefacts in numerical solutions. The main goal of this paper is to avoid any regularisation of F (u) and to decrease numerical dispersion significantly. Consequently, precise numerical solutions of () - (3) can be obtained even for the case of vanishing diffusion and for very large retardation factor F (u). 2

3 Characteristics based numerical discretisations of contaminant transport with nonlinear adsorption were described in [3, 4, 8, 5] and upwind based discretization methods for the same type of problems in [6, 22]. Asymptotic formulas for large-time behaviour of the exact solution of () were obtained in [, 7], but they can not be used directly in discretization methods. The method presented in this paper is based on operator splitting, where nonlinear transport and nonlinear diffusion are solved separately along each time step see, e.g. [2]. The transport part is solved exactly for piecewise constant initial profile and the diffusion part by some standard finite volume method. Consequently, our method has very weak restriction on time steps, and the Courant number can be larger than one. In fact, if smaller diffusion is considered (i.e., the Peclet number is larger) than larger time steps can be taken in numerical simulations. Analytical solutions for purely convective case of contaminant transport with nonlinear sorption using simple initial and boundary data were described in [26]. Independently, a general case of piecewise constant initial and boundary data was solved by different analytical methods in [6] for the same equation, where also long-time behaviours like the collapses of shocks were considered. Our paper summarises and extends results presented by authors in [6]. The paper is organised as follows. In section 2 we describe the method of solution for () - (3) using operator splitting approach. In section 3 we deal in details with purely convective transport and present analytical formulas for solution of multiple Riemann problems with Freundlich and Langmuir isotherms. In section 4 the method for solving of nonlinear diffusive transport is discussed. Finally, section 5 presents some numerical results and concluding discussions. The routines in Maple language [2] that were used for the exact solutions of nonlinear convection transport and the programme in C language for numerical solving of nonlinear convection-diffusion transport with Freundlich isotherms (using software library Numerical Recipes [25]) are available by request from the first author. 2 Operator splitting method Let τ := t j t j be a time step and u j u(x, t j ) be an approximation of the solution u(x, t) at the time level t j. The method, considered here, determines numerical solution u j in two-step procedure u j = D j (τ)t j (τ)u j, where T j (τ) represents the solution of convection part of () and D j (τ) represents the solution of diffusion part of (). More precisely, we start with solving the hyperbolic problem t F (φ) + v(x) x φ =, t (t j, t j ), (8) with the inflow boundary condition φ(, t) = C (t) and the initial condition φ(x, t j ) = u j. 3

4 Further, we define u /2 j := T j (τ)u j φ(x, t j ) (9) and solve the problem with the boundary conditions t F (φ) x (D(x) x φ) =, t (t j, t j ), () φ(, t) = C (t), x φ(l, t) =, and the initial condition φ(x, t j ) = u /2 j. Finally, we can define u j := D j (τ)u /2 j φ(x, t j ). In our practical realisation, the transport step T j (τ) is solved exactly for initial piecewise constant data and for boundary conditions of the form C (t) C j, t (t j, t j ), () and the diffusion step D j (τ) is approximated using Finite Volume Method (FVM). In the case of nonconstant velocity v(x), a transformation to new space variable y is used in both steps. In general, the result of transport step is not a piecewise constant function with respect to the fixed grid that shall be used in the diffusion step, and one has to project u /2 j to this grid. From the point of view of mass conservation, it is important that this projection is precise. In fact, we compute such projection exactly for a general type of isotherms Ψ(u). The output u j of diffusion step (realised by FVM) is again piecewise constant, so the next transport step along t (t j, t j+ ) can be realised. Thus, the projection represents the only source of numerical dispersion that can be controlled by smaller space discretization and relatively large time steps. In general, the operator splitting method, as given here, can be further improved in a straightforward way to reduce time splitting error, see, e.g., descriptions and references in [9] for Strang splitting. 3 Solution of transport problem Firstly, to solve (8) we use the transformation y = G(x) = x dz v(z) (2) and rewrite (8) into a simpler form t F (u) + y u =, t (t j, t j ), (3) 4

5 where u(y, t) = u(g(x), t) := φ(t, x). The initial condition are given by u(g(x), ) = U (x) and the boundary condition by u(, t) = C (t). The solution of (3) presented in this paper is based on theoretical results by [23, 24, 7, 8, 5] etc., which guarantee the existence, uniqueness and some properties of entropy solution for hyperbolic problems. The theoretical treatment of entropy solution has been developed for hyperbolic equations in the form t w + y f(w) =, w(y, ) = W (y) = F (U (x)) (4) that can be obtain from (3) by the transformation F (u) = w, i.e., u = f(w), where f = F is an inverse function of F. Unfortunately, in general, the function f can not be determined in an analytical form. In this section, we construct analytical solutions of (3) without having an explicit form of the function f. Our construction of the entropy solution is based on analytical solutions of a multiple Riemann problem [9, 5, 2], and it substantially depends on structural properties of F. Particularly, if F is concave (or convex) and the inverse function of /F ( ) can be expressed in an analytical form, we can construct the solution of (3) in an analytical form, too. In following Remark we list some properties of the solution of multiple Riemann problem which are sufficient for the construction of solution for (4) in an analytical form. Remark If F is concave (i.e., f is convex), then a jump in W (y) at y = y is called an acceptable shock, if W (y ) := lim y y W (y) > W (y +) := lim y y W (y). In that case, the shock at y = y Rankine-Hugoniot speed v, will move in time in y-direction with the v = f(w (y )) f(w (y +)) W (y ) W (y +) If W (y ) < W (y+), then the jump at y = y is called not acceptable shock, and it is replaced in time by rarefaction wave w, which is a smooth function of the form ( ) y y w (y y, t) = [f ], (5) t that is defined for t > and y + f (W (y )) t < y < y + f (W (y +)) t. This construction is valid until no collision between shocks and rarefaction waves appears.. 5

6 To construct the rarefaction wave function w in (5), one needs to find the inverse function of f ( ). If the inverse function of Ψ ( ) is known in an analytical form, this can be realised as we describe now. Denoting v := f (w) and using one obtains v = f (w) = F (u) = + Ψ (u), (6) ( ) u = (Ψ ) v. (7) v Using (6), (7) and w = F (u), one gets finally ( ( )) v (f ) (v) = w = F (Ψ ). (8) v If an analytical form of (Ψ ) is not known, one can solve (7) for some particular values of v numerically. If f is concave (i.e., when F is convex), then the role of acceptable and not acceptable shocks is exchanged. Particularly, if W (y ) > W (y +) then the shock is not acceptable and it develops into the rarefaction wave w. Again, the function w is generated by (5), but it has a different shape now. Summarising, the solution w of (4) for a piecewise constant initial function W consists (for some time) of constants and rarefaction waves w generated by (5). Next, we describe the construction of w in details. 3. Analytical solution of multiple Riemann problem Firstly, we simplify the treatment of boundary conditions by considering only the case C (t) W () in (2). Our description can be extended for the case when C (t) is a piecewise constant function in time, like in (), but we skip details here. Let the initial function W has a piecewise constant profile, i.e., W (y) = { W (i), for y i < y y i, i =,..., M, W (), for y y. (9) If W i W i+, the point y i is a shock. To distinguish between acceptable and not acceptable shocks, we use the notation y (a) i, resp. y (na) i. The development of shocks in time will be described by y i (t) = y i + v i t, where f (W (i) ), if y i = y (na) i v i =. f(w (i+) ) f(w (i) ), if y W (i+) W (i) i = y (a) i For a completeness, we define v =. 6

7 Clearly, for some time interval (, t ) there is no collision of two shocks, i.e. y i (t) < y i (t), i =,..., M, and the solution w(y, t) can be defined for each space interval (y i (t), y i (t)) following Remark. Particularly, if y i = y (a) i, then and if y i = y (na) i, then w(y, t) = W (i), y i (t) < y < y i (t), (2) w(y, t) = { w (y y i, t) y i (t) < y < y i + f (W (i) ) t, W (i) y i + f (W (i) ) t < y < y i (t). (2) To define the time t precisely, we discuss now possible collisions of y i (t) with y i (t). To do so, we assume first that f is convex. Note that in such a case, due to Remark, two moving shocks y i (t) and y i (t) can never meet if y i = y (na) i and y i = y (na) i. We must discuss two particular cases. Firstly, if y i = y (na) i and y i = y (a) i, then f (W (i) ) > v i, and the rarefaction wave (from left) will meet the shock (from right) at a time t i, see also figure 7 later. The time t i is given by the equation y i (t i ) = y i(t i ), i.e., t i = y i y i f (W (i) ) v i. Secondly, if y i = y (a) i and y i = y (a) i, then the shocks y i (t) and y i (t) can meet only if v i > v i, and, in such case, it happens in time t t i = y i y i v i v i. Summarising, the general solution (2) - (2) is valid for < t < t, where t is the time at which the first collision of shocks appears, i.e., t = min{t i, t i }. i To extend the description of w for t > t, we restrict to a special form of initial functions W. Definition We call the function W (y) in (9) to be concave if the values W (i) are non-decreasing for j i < M and non-increasing for i j M. To extend the description (2) and (2) for t > t, we must discuss two possible cases - collisions of several acceptable shocks and a time development of the shock after collision of acceptable and not acceptable shock. We still concentrate on the case f being a convex function, i.e., one has y i = y (na) i and y i = y (a) i, if W (i ) W (i) W (i+). 7 i,

8 Firstly, if y i and y i are acceptable shocks that met at t = t, then for t > t one has to remove the value W (i) and to identify y i (t ) with y i (t ), i.e., for t > t one has z(t) := y i (t) y i (t), where the new shock z(t) will move with the velocity v = f(w (i+) ) f(w (i ) ) W (i+) W (i ). Following Remark, one can now define the analytical solution of (3) for t (, t 2 ), t 2 > t, where the time t 2 corresponds to the collision of a rarefaction wave from the left (starting from y i ) with the moving acceptable shock z(t). In fact, z(t) can represent a shock obtained after several collisions in a decreasing leading front of W (y), i.e., with the last accumulated velocity z(t) = y i (t) =... = y i+k (t) (22) v = f(w (i) ) f(w (i+k) ) W (i) W (i+k). In such case, the time t 2 can be determined from the equation for t y i + f (W (i) )t = y i + t v(s)ds, (23) where v(s) is the shock velocity of y i that is changing by jumps after each collision. In general, there is no particular order of the collisions of neighbouring acceptable shocks y k and y k for different k with k i. Finally, to construct the solution of (3) for t > t 2, one has to describe the movement of the shock at the front of rarefaction wave. The point s(t 2 ) = y i + f (W (i) ) t 2 represents the acceptable shock at t = t 2 that will move further for t > t 2 with the Rankine - Hugoniot velocity ṡ(t) that is governed by the ODE ṡ(t) = f(w (s(t) y i, t)) f(w (i+k) ) (24) w (s(t) y i, t) W (i+k) and the initial condition s(t 2 ) = y i + f (W (i) ) t 2. (25) The equation (24) is valid up to the next collision with acceptable shock or up to the moment when w (s(t) y i, t) = W (i+k) which is a singular point for (24). Nevertheless, in both cases we are in a situation which was described before. The case of f being a concave function can be treated analogously. Next, we describe in details the application of our construction for the case of Freundlich and Langmuir type of isotherms. 8

9 y y Figure : The left picture - the exact solution of t (u + u /2 ) + y u = for t = (i.e., the initial function U ), and for t = 7.4. The right picture - the corresponding rarefaction wave u (y, 7.4) defined in (27). 3.2 Freundlich isotherms In this case, F (u) = u + au p with p >. If p <, then F is concave, and, consequently, f is convex. For p >, f is concave, so the type of shocks in W is different. Therefore, we consider each case separately Case p (, ) Generally, one can not express f in an analytical form. Nevertheless, we can construct the rarefaction wave by using (8), ( ( ) ) v w = [f ] p (v) = F. p av Following (5), the corresponding rarefaction wave for (4) is of the form ( (t ) ) y w p (y, t) = F, < y < t. (26) p ay Consequently, the corresponding rarefaction wave for (3) is u (y, t) = ( ) t y p, < y < t. (27) p ay Concerning the evolution of initial profile W, the solution w(y, t) takes the form (2) for all y i = y (na) i, i < i, and the form (2) otherwise, i.e., for all y i = y (a) i, i i. The pictures in figure illustrate the construction (2) and (2) to determine u using a single rarefaction wave function u defined by (27). This construction is valid until the rarefaction wave does not meet a shock, see Section 3. for details. The governing equation for the shock propagation 9

10 after the rarefaction wave meets the acceptable shock z(t) from (22) is governed by the ODE (24) for t > t 2, i.e., where ṡ(t) = A(s(t), t) f(w (i+k) ) F (A(s(t), t)) W (i+k), s(t 2) = y i + f (W (i) ) t 2, (28) ( ) t s(t) + yi p A(s(t), t) :=. p a(s(t) y i ) In general, this ODE must be solved numerically. For some particular values of p, the function s(t) can be obtained by solving a nonlinear algebraic equation, see [6] for some examples. To characterise the solutions of transport equation with Freundlich isotherm, we present a representative example of (3). Particularly, we consider the zero inflow concentration C (t) and the piecewise constant initial impulse U (y) = for y (, ) and U (y) = otherwise. The following figures were produced using the software Maple [2], and the corresponding Maple routines are available by request from the first author..8.6 u y Figure 2: Solutions of t (u + u p ) + y u = for p = 3/4, /2, /4 (from the right to the left - the gray, dash and solid lines) and for three different times t = 3, 6, 4. The figure 2 shows the time development of solutions of the equation t (u + u p ) + y u = for three different values of parameter p (, ). In each case, the rarefaction wave starts at the point y = y = (in fact, y (t) ), and the tailing part of rarefaction wave appears behind the acceptable shock that has started initially at y =. For smaller values of p the rarefaction wave reaches sooner the shock at the front of curves and the decrease of shock value afterwards is faster. Of course, with smaller values of p the retardation of transport (7)

11 is larger, and, consequently, the movement of fronts (or acceptable shocks) is slower. The figure 3 shows the curves F (u), i.e., the solution w of (4), for the same data. Note that the solutions w = F (u) are mass conservative. 2.5 w y Figure 3: The curves w = u + u p for solutions of t (u + u p ) + y u = for p = 3/4, /2, /4 (from the right to the left - the gray, dash and solid lines) and for three different times t = 3, 6, Case p > The governing equations are the same as in the case p <. Particularly, the rarefaction waves w (y, t) and u (y, t) are given by the same formulas (26) and (27). Opposite to the previous case, one has f (W (i ) ) f (W (i) ), and acceptable shocks appear on the trailing part of solution curves (i.e., at the points y i, i < i ), whereas the rarefaction waves appear on the leading front of initial concave profile W (i.e., at the points y i, i i, see the Definition ). Also the governing equation for the movement of shock after its collision with the rarefaction wave is given by an analogous ODE to (28), where one has only to replace k by k, since the shocks can accumulate on the trailing part of wave. The solution of equation t (u + u p ) + y u = with the same initial and boundary conditions as in the case p < are presented in next figures. The figure 4 shows u for three different values of p (, 2) where the (convex) shape of curves is analogous to the case p (, ). Of course, the tailing part of rarefaction waves appears in the front of curves. For p [2, 4], the behaviour is analogous, but the shape of curves becomes concave for larger p.

12 Figure 4: Solutions of t (u + u p ) + y u = for p = 7/4, 3/2, 5/4 (from the left to the right - the gray, dash and solid line) for three different times t = 4, 8, Figure 5: Solutions of t (u + u p ) + y u = for p = 2, 3, 4 (from the left to the right - the solid, dash and gray line) for three different times t = 3, 9, 8. 2

13 3.2.3 Case p =. In this case, all shocks of initial profile are acceptable and the construction of analytical solution is trivial. 3.3 Langmuir isotherm In this case, the function F (s) = as/( + bs) is concave, and, consequently, f is convex. One can express f directly in the analytical form f(w) = ( w a + + ) (bw + a)2 + 4a, 2 b b and ( [f ] (v) = a + 2 ) 2v a. b (2v ) 2 Thus, the rarefaction wave w is of the form ( w (y, t) = a + 2 ) 2y t a, (29) b t2 (2y t) 2 and, consequently, ( u (y, t) = + ) 2y a. (3) b t2 (2y t) 2 The construction of analytical solution for concave initial profile W is analogous to the Case I of Freundlich isotherm with p <. Of course, opposite to the Freundlich case, one has F () = +a <. Particularly, the velocity speed of rarefaction wave between y (t) up to the shock y (t) (provided C (t) ) is f () = ( + a) >, opposite to the Freundlich case p <, where y (t). Consequently, the trailing part of concentration profile is shorter for the Langmuir case. When the rarefaction wave meets the shock in front, i.e., at t = t 2, the governing equation for the shock movement is identical to (28), where one has to use A(s(t), t) := u (s(t) y i, t). For the initial profile W (y) of the form W (y) W, y (, L), and W (y) otherwise, one can find an explicit formula for the front movement, s(t) = 2(a + ) 2 ( 4ta + 2P b(a ) 4t a ( + P b t ) ( a P b ) ) t for P t > t = f (W )W f(w ), where P = W L. The solutions of equation t (u + au/( + u)) + y u = for three different values of a, with the identical boundary and initial conditions as before, are presented in figure 6. 3

14 y Figure 6: Solutions of t (u + au/( + u)) + y u = for a = 2.83, 6, (from the left to the right - the solid, dash and gray line) and for three different times t = 5.83, 25, General concave and convex isotherms Ψ For a general case of isotherm Ψ that is either convex or concave, the inverse function of Ψ needs not to be known in an analytical form. Nevertheless, one can construct the rarefaction wave for some finite set of points numerically by solving a nonlinear algebraic equation. Particularly, to obtain u (y y, t), one can follow the approach in (7) and solve the following nonlinear scalar algebraic equation Ψ (u (y y, t)) = t y + y y y. with respect to u by means of, for instance, iterative Newton method. In the case when an isotherm is neither convex nor concave, one can numerically compute the rarefaction waves following the Osher s solution, see, e.g., [9] for details. 4 Solution of nonlinear diffusion problem In previous section we have derived analytical formulas for the solution of nonlinear convection equation (3) with piecewise constant initial function that was defined on some grid = y < y <... < y M = G(L). It is more convenient to transform the nonlinear diffusion equation () to y-variable, too. It means, in the second step of operator splitting method we have to solve the equation t F (u) a(y) y (b(y, t) y u) =, (3) 4

15 with the boundary condition and the initial condition where the coefficients a and b are given by u(, t) = C (t), y u(g(l), t) =, (32) u(y, t j ) = u /2 j (y), (33) a(y) := v(g (x)), b(y) := D(G (x)) v(g (x)). The function u /2 j (y) in (33) is an output of the first transport step in the operator splitting procedure (9), see Section 2. Following a standard FVM strategy to approximate (3), one obtains τa i ( h i F (u j i ) yi b i u j i+ uj i h i++h i 2 y i F (u(y, t j )) dy b i u j i uj i h i+h i 2 ) = (34) for i =,..., M, where h i = y i y i and a i := a(y i h i /2) and b i = b(y i ). Note that u j i u(y i h i /2, t j ). Due to the boundary conditions, we complete u j = C (t j ) and u j M+ = uj M. The resulting nonlinear algebraic system can be solved by standard Newton iterative method using analytical linearisation. Another possibility is to use linearisation by relaxation scheme that was developed in [3, 4] and has been justified in papers [2, 2,, ], etc., see [6] for more details. The crucial point in the finite volume discretization method of (3) (that represents in a discrete form the mass balance property of origin analytical equation) is the evaluation of the second integral in (34). Our aim is to compute such integrals exactly, see next section. 4. Exact evaluation of integral in (34) The function u(x, t j ) in (34) is given by (33), so one has to compute yi y i F (u /2 j (y)) dy. (35) The function u /2 j consists of constants and rarefaction waves. The constant parts, although they do not fit to computational grids, make no difficulties in computations of (35). In the case of rarefaction waves (that are in general not available in an analytical form), an application of numerical quadrature for integral in (35) can lead to mass balance violations and solution errors. 5

16 To avoid such errors, we propose a procedure to evaluate the integral in (35) exactly by evaluating the rarefaction waves only in some finite number of discrete points. Note that w = F (u). Let us denote J(α, β; t ) := β α w (y, t ) dy. (36) The function w in (36) represents the rarefaction wave that starts at t = as a constant function with not acceptable shock y (na) located at time t = in origin, i.e., y (na) () =. The position of this shock at t = t will be denoted by ξ := y (na) (t ) and is given for isotherms (4) - (6) by p < (Freundlich and the mixed type of isotherm), ξ = t p > (Freundlich and the mixed type of isotherm), +a t Langmuir type of isotherm. We consider first the case ξ < α < β, the case α < β < ξ (that occurs for Freundlich isotherms with p > ) will be treated afterwards. To evaluate J(α, β; t ), one can first compute the values J(ξ, α; t ) and J(ξ, β; t ) and then use J(α, β; t ) = J(ξ, β; t ) J(ξ, α; t ). To compute J(ξ, β; t ), we track the acceptable shock y (a) (t ) = β backward in time until t = and we denote its initial (not yet known) position by h := (). Clearly, a constant function w(y, ) that has the nonzero value w(y, ) A := w (β, t ) only in interval (, h) will be transported to w (y, t ) at t = t, see figure 7 for an illustration. As the variable w is mass conservative, one obtains J(ξ, β; t ) = A h. To determine the value A h, one can use the fact that the rarefaction wave meets the front shock at t = t, i.e., y (a) From (37) one obtains that Hence t f (A) = β, Ah = A(β (β h)) = t (Af (A) f(a)) = β t f(a) A = β h. (37) ( A f(a) ) f. (A) J(ξ, β; t ) = β (F (u (β, t )) u (β, t )F (u (β, t ))). (38) Finally, for the case that β < ξ, we have to evaluate J(β, ξ; t ), see also the figure 8. One has instead of (37) that t f (A) = β, 6 t f(a) A = β + h, (39)

17 y Figure 7: Constant function of value A = 2 at t = with h = that develops to rarefaction wave w (β, t ) at t = 6 with ξ = and β = 4 for the case of Freundlich isotherm with p = /2. The both areas are identical y Figure 8: Constant function of value A = 2 at t = with h = that develops to rarefaction wave w (β, t ) at t = with β = 4 and ξ = for the case of Freundlich isotherm with p = 3/2. The both areas are identical. 7

18 and one obtains Ah = A(β + h β) = t (f(a) Af (A)) = β ( ) f(a) f (A) A. As J(β, ξ; t ) = J(ξ, β; t ), the formula (38) is valid also for this case. Summarising, to compute the integral in (35) exactly, one needs to determine the values of rarefaction waves only for some finite set of points β and to apply the formula (38). If no analytical form of rarefaction wave is known, one can find the values A = w (β, t ) = F (u (β, t )) by solving the equation f (A) = β/t with respect to A numerically, see also section Numerical experiments In this section we describe numerical experiments that confirm an applicability of described methods for transport problems with nonlinear sorption. First, the exact solutions for the equation (if d = ) t F (u) + x u d xx u =, (4) u(, x) =, x (, ), u(, x) =, x / (, ), u(t, ) =, t > for F given by (4) and (5) were implemented using the package Maple 7. [2], see the figures in previous sections. Such solutions can be constructed directly using definitions (7) if t (, t ), where t is a time when a rarefaction wave meets the shock in front. For the case when the speed of shock must be determined solving the ODE (24), built-in Maple procedures can be used to obtain very precise numerical solutions. Finally, to solve the general form of (4) with d numerically, the operator splitting method was implemented as described in the section 2. The transport part was solved following the section 4. and the discretisation of diffusive part followed the section 4.2 with an analytical linearisation used in Newton method. Extensive tests were done for the most interesting (and the most complicated case) of Freundlich isotherm with p <. As F () =, a modification of analytical linearisation used in Newton method must be done. We avoided a modification (a regularisation ) of the function F itself. To modify F (u) only, a small parameter ɛ (to be set by the user) was chosen and F (u) was replaced by F (u), where F (u) := { F (u), if u ɛ F (ɛ), if u < ɛ. (4) The implementation of iterative procedure for Newton method was realised using C-language procedures of Numerical Recipes [25]. 8

19 x x x Figure 9: Numerical solutions of t (u + u 3/4 ) + x u d xx u = for d = 2, 3, 4 at t = 3 (from the left to the right) that are compared with the exact solution for d = x x x Figure : Numerical solutions of t (u + u 3/4 ) + x u d xx u = for d = 2, 3, 4 at t = 5 (from the left to the right) that are compared with the exact solution for d =. To illustrate that the method described in this paper can be applied for d in (4), the following examples were realised. The grid step in all examples was chosen to be uniform h i h =.5. Firstly, the system (4) - (4) was computed for a smaller time (t = 3) before the rarefaction wave meets the shock in front. These computation were realised for d = 2, 3 and 4, see the figure 9, where the numerical solution is compared with the exact solution for d = to judge the influence of nonzero physical diffusion. All computations were realised using 2 time step, e.g., τ =.5 and Courant number was approximately 7. Clearly, the method produces for different d different results that is not the case for many numerical methods when solving (4) with very small diffusion. Secondly, analogous tests were done for larger times with rarefaction wave behind the shock in front. Again, see figure, the method can produce clearly different results for different small diffusion coefficients. In all computations τ =.5 with the Courant numbers varying around the value 7. 9

20 6 Conclusions In this paper semi-analytical solutions for one-dimensional problems of contaminant transport with nonlinear adsorption were constructed. The method is based on operator splitting where the convective transport is solved exactly for piecewise constant initial condition and the diffusive transport is solved by finite volume method. For small diffusion coefficients, the advantage of this method is that very large time steps can be used to obtain precise numerical results. Such semi-analytical solutions can be utilised as benchmark examples for numerical packages. Moreover, the solution procedure presented here can be used for inverse problems that are dealing with determination of soil properties. References [] Babušikova, J.: 2, Application of relaxation scheme to degenerate variational inequalities. Applications of Mathematics 46, [2] Bänsch, E.: 995, Numerical experiments with adaptivity for the porous medium equation. Acta Math.Univ.Comenian.(N.S.) 64, [3] Barrett, J., H. Kappmeier, and P. Knabner: 994, Lagrange-Galerkin approximation for advection-dominated nonlinear contaminant transport in porous media. In: Computational methods in water resources X., Volume. pp [4] Barrett, J., H. Kappmeier, and P. Knabner: 997, Lagrange-Galerkin approximation for advection-dominated contaminant transport with nonlinear equilibrium or non-equilibrium adsorption. In: R. H. et. al. (ed.): Modelling and Computations in Environmental Science. pp [5] Dafermos, C.: 972, Polygonal approximation of solutions of the initial value problem for a conservation law. J.Math.Anal.Appl. 38, [6] Dawson, C.: 998, Analysis of an upwind - mixed finite element method for nonlinear contaminant transport equations. SIAM J.Numer.Anal. 35(5), [7] Dawson, C., C. Van Duijn, and R. Grundy: 996, Large time asymptotics in contaminant transport in porous media. SIAM J.Appl. Math. 56, [8] Dawson, C., C. Van Duijn, and M. Wheeler: 994, Characteristic-Galerkin methods for contaminant transport with non-equilibrium adsorption kinetics. SIAM J.Numer.Anal. 3(4), [9] Frolkovič, P. and J. Geiser: 2, Numerical simulations of radionuclides transport in double porosity media with sorption. In: A. H. et al. (ed.): Algoritmy 2. pp

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