I am a newbie in Julia an I have written the following code:

using LinearAlgebra
using Plots

    t = 0;

    epsilon = 0.1;
    

    ########## initial data: Sod problem (rho,u,T) ##########
    
    Prim_l = [1 0 1];                           # 0 <= x <= 0.5
    Prim_r = [0.125 0 0.1];                     # 0.5 < x <= 1

    ########## domain in space ##########

    x_start = 0;                                # start
    x_end   = 1;                                # end

    k   = 250;                                  # cell discretization, grid-points
    dx  = (x_end-x_start)/k;

    ########## velocity space (v = molecular velocities at certain point x at time t) ##########

    v_start = -4.5;                             # start
    v_end   = 4.5;                              # end

    v_nodes = 100;                              # velocity discretization, velocity-points

    v = collect(range(v_start,v_end,v_nodes));  # collect(range(a,b,n)) <-> LinRange(x): works! 
    w = 9/v_nodes*ones(size(v));                # width (for numerical integration)

    ########## calculation initial elements of U ##########
   
    xx = LinRange(x_start,x_end,k+1);
    xm = 0.5*(xx[1:end-1] + xx[2:end]);

    U       = zeros(k,3);
    g_val   = zeros(k+1,v_nodes);

    U[:,1] = (xm.<=0.5)*Prim_l[1]+(xm.>0.5)*Prim_r[1];      # rho
    U[:,2] = (xm.<=0.5)*Prim_l[2]+(xm.>0.5)*Prim_r[2];      # u bulk velocity of the gas
    U[:,3] = (xm.<=0.5)*Prim_l[3]+(xm.>0.5)*Prim_r[3];      # temperatur

    ########## initial data: Flux, Source ##########

    Flux    = zeros(k+1,3);    
    Source  = zeros(k,3);

    ########## parts of v ##########

    vvp     = kron(ones(k+1,1),v'.*(v'.>0));    # v^{+}
    vvm     = kron(ones(k+1,1),v'.*(v'.<0));    # v^{-}
    vv      = kron(ones(k+1,1),v');             # v
    vv_s    = kron(ones(k,1),v');

    dt      = 0.0005;                           # time-step
    t_end   = 0.14;                             # running time

    while t < t_end
        
        M           = Maxwell(U, v);
        vM          = M[[1;1:k],:].*vvp + M[[1:k;k],:].*vvm;                                                        # term: <m*(v^{+}*M^{n}_{i} + v^{-}*M^{n}_{i+1})>
        
        vg_dx       = (g_val[1:k+1,:]-g_val[[1;1:k],:]).*vvp/dx + (g_val[[2:k+1;k+1],:]-g_val[1:k+1,:]).*vvm/dx;    # term: [v^{+}*(g^{n}_{i+1/2}-g^{n}_{i-1/2})/deltax + v^{-}*(g^{n}_{i+3/2}-g^{n}_{i+1/2})/deltax]
        vM_dx       = vv.*(M[[1:k;k],:]-M[[1;1:k],:])/dx;                                                           # term: (v*(M^{n}_{i+1}-M^{n}_{i})/deltax)

        A           = id_min_proj(U, M, vg_dx, vM_dx, v, w);                        # return value id_min_proj: tensor([251,100],2)
        vg_dx       = A[1];
        vM_dx       = A[2];

        g_val       = (g_val - dt*vg_dx - dt/epsilon*vM_dx)/(1+dt/epsilon);         # kinetic equation

        Flux[:,1]   = vM*w;
        Flux[:,2]   = (vv.*vM)*w;
        Flux[:,3]   = 0.5*(vv.^2 .*vM)*w;
        
        g_dx        = (g_val[2:k+1,:]-g_val[1:k,:])/dx;
        Source[:,1] = g_dx*diagm(w)*v;
        Source[:,2] = (vv_s.*g_dx)*diagm(w)*v;
        Source[:,3] = 0.5*(vv_s.^2 .*g_dx)*diagm(w)*v;

        U           = U - dt/dx*(Flux[2:end,:]-Flux[1:end-1,:]) - dt*epsilon*Source;        # macroscopic equation

        t           = t+dt;

    end

    rho = U[:,1];
    u   = U[:,2]./rho;
    T   = 2*U[:,3]./rho-u.^2;

I get the following output in REPL:

id_min_proj (generic function with 1 method)

Maxwell (generic function with 1 method)

┌ Warning: Assignment to `g_val` in soft scope is ambiguous because a global variable by the same name exists: `g_val` will be treated as a new local. Disambiguate by using `local g_val` to suppress this warning or `global g_val` to assign to the existing global variable.
└ @ d:ProgrammierungJulia.ProjekteTest.jl:71
┌ Warning: Assignment to `U` in soft scope is ambiguous because a global variable by the same name exists: `U` will be treated as a new local. Disambiguate by using `local U` to suppress this warning or `global U` to assign to the existing global variable.
└ @ d:ProgrammierungJulia.ProjekteTest.jl:82
┌ Warning: Assignment to `t` in soft scope is ambiguous because a global variable by the same name exists: `t` will be treated as a new local. Disambiguate by using `local t` to suppress this warning or `global t` to assign to the existing global variable.
└ @ d:ProgrammierungJulia.ProjekteTest.jl:84
ERROR: UndefVarError: `U` not defined in local scope
Suggestion: check for an assignment to a local variable that shadows a global of the same name.
Stacktrace:
 [1] top-level scope
   @ d:ProgrammierungJulia.ProjekteTest.jl:61

julia> M
ERROR: UndefVarError: `M` not defined in `Main`
Suggestion: check for spelling errors or missing imports.

Okay … the matrix M is not calculated, I think, due to the warnings.
I took some looks at books about Julia with code examples. I cannot find any hints with this scopes.
All in all … this code works in Matlab. I have re-write this code in Julia, with some changes (such as the matrix-notations).

Kind regars!

Julia documentation, books about Julia and code examples