n=10;

Δz=L/n;
eq1=Table[(cD1/((Δz)^2))(Subscript[u,j+1][t]+Subscript[u,j-1][t]-2Subscript[u,j][t])-tr Subscript[u,j][t]Subscript[w,j][t]==D[Subscript[u,j][t],t],{j,1,n-1}];
eq2=Table[(cD2/((Δz)^2))(Subscript[w,j+1][t]+Subscript[w,j-1][t]-2Subscript[w,j][t])-tr Subscript[u,j][t]Subscript[w,j][t]==D[Subscript[w,j][t],t],{j,1,n-1}];
eq3=Table[(cD3/((Δz)^2))(Subscript[v,j+1][t]+Subscript[v,j-1][t]-2Subscript[v,j][t])+tr Subscript[u,j][t]Subscript[w,j][t]==D[Subscript[v,j][t],t],{j,1,n-1}];

eq3

cf1={Subscript[u,n][t]->Subscript[u,n-1][t],Subscript[u,0][t]->Subscript[u,1][t]+((Δz)/cD1)Jz1};
cf2={Subscript[w,0][t]->Subscript[w,1][t],Subscript[w,n][t]->Subscript[w,n-1][t]+((Δz)/cD2)Jz2};
cf3={Subscript[v,0][t]->Subscript[v,1][t],Subscript[v,n][t]->Subscript[v,n-1][t]};

cf3

cfs=Join[cf1,cf2,cf3]

eqs=Join[eq1,eq2,eq3]/.cfs;

Dimensions[eqs]

ci1=Table[Subscript[u,j][0]==0,{j,1,n-1}];
ci2=Table[Subscript[w,j][0]==0,{j,1,n-1}];
ci3=Table[Subscript[v,j][0]==0,{j,1,n-1}];
cis=Join[ci1,ci2,ci3];

cis

difeq=Join[eqs,cis];

difeq

conds={u0->1,L->1,cD1->0.1,cD2->0.1,cD3->0.01,tr->100,Jz1->0.1,Jz2->0.1};
ndifeq=difeq/.conds;

ndifeq

var1=Table[Subscript[u,j][t],{j,1,n-1}];
var2=Table[Subscript[w,j][t],{j,1,n-1}];
var3=Table[Subscript[v,j][t],{j,1,n-1}];
var=Join[var1,var2,var3];

var

sol=NDSolve[ndifeq,var,{t,0,200}];

t0=0.1;lpu=Table[{(j Δz/.conds),(Subscript[u,j][t]/.cf1/.sol[[1]]/.{t->t0})},{j,0,n}];
lpw=Table[{(j Δz/.conds),(Subscript[w,j][t]/.cf2/.sol[[1]]/.{t->t0})},{j,0,n}];
lpv=Table[{(j Δz/.conds),(Subscript[v,j][t]/.cf3/.sol[[1]]/.{t->t0})},{j,0,n}];

ListPlot[{lpu,lpw,lpv},Joined->False,PlotRange->All]

n=100;

Δz=L/n;
eq1=Table[(cD1/((Δz)^2))(Subscript[u,j+1][t]+Subscript[u,j-1][t]-2Subscript[u,j][t])-tr Subscript[u,j][t]Subscript[w,j][t]==D[Subscript[u,j][t],t],{j,1,n-1}];
eq2=Table[(cD2/((Δz)^2))(Subscript[w,j+1][t]+Subscript[w,j-1][t]-2Subscript[w,j][t])-tr Subscript[u,j][t]Subscript[w,j][t]==D[Subscript[w,j][t],t],{j,1,n-1}];
eq3=Table[(cD3/((Δz)^2))(Subscript[v,j+1][t]+Subscript[v,j-1][t]-2Subscript[v,j][t])+tr Subscript[u,j][t]Subscript[w,j][t]==D[Subscript[v,j][t],t],{j,1,n-1}];
cf1={Subscript[u,n][t]->Subscript[u,n-1][t],Subscript[u,0][t]->Subscript[u,1][t]+((Δz)/cD1)Jz1 ((Sin[2π t])^2)};
cf2={Subscript[w,0][t]->Subscript[w,1][t],Subscript[w,n][t]->Subscript[w,n-1][t]+((Δz)/cD2)Jz2};
cf3={Subscript[v,0][t]->Subscript[v,1][t],Subscript[v,n][t]->Subscript[v,n-1][t]};
cfs=Join[cf1,cf2,cf3];
eqs=Join[eq1,eq2,eq3]/.cfs;
ci1=Table[Subscript[u,j][0]==0,{j,1,n-1}];
ci2=Table[Subscript[w,j][0]==0,{j,1,n-1}];
ci3=Table[Subscript[v,j][0]==0,{j,1,n-1}];
cis=Join[ci1,ci2,ci3];
difeq=Join[eqs,cis];
conds={u0->1,L->1,cD1->0.2,cD2->0.2,cD3->0.1,tr->100,Jz1->0.1,Jz2->0.1};
ndifeq=difeq/.conds;
var1=Table[Subscript[u,j][t],{j,1,n-1}];
var2=Table[Subscript[w,j][t],{j,1,n-1}];
var3=Table[Subscript[v,j][t],{j,1,n-1}];
var=Join[var1,var2,var3];
sol=NDSolve[ndifeq,var,{t,0,200}];

t0=5;lpu=Table[{(j Δz/.conds),(Subscript[u,j][t]/.cf1/.conds/.sol[[1]]/.{t->t0})},{j,0,n}];
lpw=Table[{(j Δz/.conds),(Subscript[w,j][t]/.cf2/.conds/.sol[[1]]/.{t->t0})},{j,0,n}];
lpv=Table[{(j Δz/.conds),(Subscript[v,j][t]/.cf3/.sol[[1]]/.{t->t0})},{j,0,n}];
ListPlot[{lpu,lpw,lpv},Joined->True,PlotRange->{-0.05,1.05},Filling->{{1->0,2->0,3->0}}]

ListPlot[lpv]

tf=3.0;
zm=1.0;
plotu={};
plotw={};
plotv={};
ProgressIndicator[Dynamic[prog]]
Do[
prog=vz/zm;
Do[
list=Table[{(j Δz/.conds),(Subscript[u,j][t]/.cfs/.conds/.sol[[1]]/.{t->vt})},{j,0,n}];
vu=Interpolation[list,vz];
AppendTo[plotu,{vt,vz,vu}];
list=Table[{(j Δz/.conds),(Subscript[w,j][t]/.cfs/.conds/.sol[[1]]/.{t->vt})},{j,0,n}];
vw=Interpolation[list,vz];
AppendTo[plotw,{vt,vz,vw}];

list=Table[{(j Δz/.conds),(Subscript[v,j][t]/.cfs/.conds/.sol[[1]]/.{t->vt})},{j,0,n}];
vv=Interpolation[list,vz];
AppendTo[plotv,{vt,vz,vv}];

,{vt,0,tf,tf/40}];
,{vz,0,zm,zm/40}]

ListPlot3D[{plotu,plotw,plotv},PlotStyle->{Opacity[0.5],Opacity[0.5],Opacity[1]},AxesLabel->{"t","z","Cons"},Mesh->None]