FDM for TWI

TWIPlot[Q1_, Q2_, Q3_]  :=  
      Show[
ListPlot[
          Abs[Q1], 
PlotStyle -> 
            RGBColor[1, 0, 0], 
PlotJoined -> 
            True, 
DisplayFunction -> 
            Identity
], 
ListPlot[
          Abs[Q2], 
PlotStyle -> 
            RGBColor[0, 1, 0], 
DisplayFunction -> 
            Identity, 
PlotJoined -> 
            True
], 
ListPlot[
          Abs[Q3], 
PlotStyle -> 
            RGBColor[0, 0, 1], 
DisplayFunction -> 
            Identity, 
PlotJoined -> 
            True], 
DisplayFunction -> $DisplayFunction, 
PlotRange -> {0, 
            3}
]

Clear[Q10]
Δx = 0.1;
Q10  =  
        Table[(E^(-(x^2))),  {x,  -10, 
            10, Δx}];
Q20  =  
        Table[(E^(-(((x - 2))^2))),  {x,  -10, 
            10, Δx}];
Q30  =  
        Table[2 (E^(-(((x + 2))^2))),  {x,  -10, 
            10, Δx}];
TWIPlot[ Q10, Q20, Q30]

TimeStep[{Q1In_List, Q2In_List, 
          Q3In_List},  Δx_, Δt_] :=  
      Module[
{Blah}, 
{γ1, γ2, γ3} = {1, -1, 1}; 
{c1, c2, 
            c3}  =  {1, 2, 
            3}; 
{
Q1In  - 
            c1 Δt/(2 Δx) ( 
                RotateLeft[Q1In, 1] - 
                  RotateRight[Q1In, 
                    1])  +  γ1 Δt Conjugate[Q2In] 
              Conjugate[Q3In], 
Q2In  - 
            c2 Δt/(2 Δx) ( 
                RotateLeft[Q2In, 1] - 
                  RotateRight[Q2In, 
                    1])  +  γ2 Δt Conjugate[Q1In] 
              Conjugate[Q3In], 
Q3In  - 
            c3 Δt/(2 Δx) ( 
                RotateLeft[Q3In, 1] - 
                  RotateRight[Q3In, 
                    1])  +  γ3 Δt Conjugate[Q1In] 
              Conjugate[
                Q2In]
}
]

{Q1, Q2, Q3}  =  {Q10, Q20, Q30};
Δt = 0.01;
Table[
{Q1, Q2, Q3}  =  
        TimeStep[{Q1, Q2, 
            Q3}, Δx, Δt]; 
TWIPlot[ Q1, Q2, 
        Q3], 
{200}]