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Appendix A Calculus of eigenvalues and eigenvectors The eigenvalues of matrix e Jnare the roots of the following polynomial: (e λ3+a1e λ2+a2e λ+a3) = 0 (A.1) whose coefficients a1,a2and a3are given by: a1=−2eu a2=−pbz e B−ec2+eu2a3=−pbz e A(A.2) Calling Q= (3a2−a2 1)/9 and R= (9a1a2−27a3−2a3 1)/54, the eigenvalues are real if Q3+R2<0, and are given by e λ2= 2√−Qcos(θp/3) −a1/3 e λ3= 2√−Qcos((θp+ 2π)/3) −a1/3 e λ4= 2√−Qcos((θp−2π)/3) −a1/3 (A.3) with θp= arccos(R/p−Q3). The eigenvectors e em,m= 1,2,3 associated to e λm, m= 1,2,3 are given by e em=   1 e λm −ec2+eu2+ e λm( e λm−2eu) pbz   (A.4) Appendix B Conservation of the numerical scheme For the sake of clarity the necessity of the last term in the numerical scheme, B.1, is going to be explained. The lack of this term in the computed method may lead to a non conservative solution, which has been misunderstood by some authors as a diffusivity problem, Un+1 i=Un i− NE X k=1 4 X m=1 (e λ−α−β−)m ke em JI,klk ∆t Ai− NE X k=1 δEIi,knklk ∆t Ai (B.1) In case of having a set of equations ∂U ∂t +∂F(U) ∂x +∂G(U) ∂y =S(U,x,y) (B.2) which can be manipulated as follows ∂U ∂t +Mn∂U ∂x +∂U ∂y −Hn∂U ∂x +∂U ∂y =Ss(U,x,y) (B.3) where Mnis the flux normal to a direction given by the unit vector n,En =Fnx+Gny, defined as Mn=∂(En) ∂U(B.4) and Hnis the flux associated to the bed slope, projected onto the unit vector n, Tbn=Sbnx+Sbny Hn=∂(Tbn) ∂U(B.5) It is possible to define the following Jacobian matrix, Jn, through the definitions in (B.4) and (B.5) 80 Conservation of the numerical scheme Jn=Mn−Hn(B.6) which will allow us to define system (B.2) as belonging to the family of hyperbolic systems. ∂U ∂t +Jn∂U ∂x +∂U ∂y =Ss(U,x,y) (B.7) Due to the non linearity of the flow En, the Jacobian matrix has to be approximated in order to generate a local linearization. Roe (1986), proposed an approximated Jacobian matrix, e Jn,k, for clean water, imposing that e Jn(Ui,Ui) = Jn(Ui) (B.8) In Murillo & Garc´ıa-Navarro (2010) an augmented Roe solver is proposed to allow the inclusion of the sediment transport terms. Hence, it was developed the building of an approximate Jacobian matrix e Jn,k at each kedge of each cell combining the normal flux En=Fnx+Gnywith the bed slope source term Tn,b at each cell edge, (δE−Tb)knk= (f Mn−e Hn)k(Uj−Ui) (B.9) (δE−Tb)knk=e Jn,kδUk(B.10) with δ(En)k= (Ej−Ei)nk,δUk=Un j−Un i, and Un iand Un jthe initial values at cells iand jsharing edge k. Figure B.1 collects the previous information in order to solve the Riemann problem in a 2D situation. Un iUn j U Un i Un j nk x0 x0 x0= 0 Figure B.1: Riemann problem in 2D along the normal direction to a cell side 81 On the other hand, the main difficulties in the definition of the approximate Jacobian matrix (B.9) for sediment transport that allows the construction of approximate solutions arises from the presence of a local Agvalue, which is variable within each cell, see Figure B.2. For this reason the net exchange of flow between internal walls must include the difference between the value of the cell in its centroid, Ag,i, and the value close to the wall within the cell Ag,I. This fact leads to new definitions of Eat each pair of cells iand j, connected through edge k, that will be referred to as EI=E(Ui, Ag,k)EJ=E(Uj, Ag,k) (B.11) An g,I,k An g,J,k An g,i An g,j Figure B.2: Linear representation by cells. Analyzing the flux term δEn drives to δEn =Ej−Ei=EJ−EI+ (Ej−EJ)−(Ei−EI) = (B.12) =δEJI +δEjJ −δEiI = = (δEJI +δEIi)−δEJj = = (δE− JI +δEIi) + δE+ JI −δEJj = = (δE− JI +δEIi)−(δE− IJ +δEJj) As it can be appreciated not only a flux crossing the wall, δE− JI , is necessary to ensure a conservative numerical scheme. A flux with information of the variation of AgI,i is also necessary and this is the additional term which appears in the numerical scheme.