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In the literature it is easy to find books that show, step by step, how to get the solution of partial differential equation, but it is not so easy to do the same for PDE systems.

Let's say we have first order, strictly hyperbolic, nonlinear, coupled 2x2 system of pdes, in one dimension, that looks like this: $$ \begin{cases} u_t+f(u,v) u_x=h(u,v) \\[2ex] v_t+g(u,v) v_x=m(u,v) \\[2ex] (u(x,0),v(x,0))=(u_0 (x), v_0 (x)) \end{cases} $$ (This is a Cauchy problem). If $(u_0 (x), v_0 (x))$ is discontinuous, e.g. $$(u(x,0),v(x,0))= \begin{cases} (u_l , v_l), x<0 \\[2ex] (u_r , v_r), x\geq0 \end{cases}$$ we would talk about Riemann problem.

I want to solve this problem using method of characteristics.

My question is: Is there any book/note/paper where I could find a step by step solution of quasilinear system that looks like this by using method of characteristics? Or maybe an easier quasilinear system (e.g. if we put right hand sides equal zero, put $f$ equal to $g$,...). It doesn't matter if it is a Riemann or Cauchy problem. I just want to see how system of this type could be solved in detail. It could be shown on any concrete system (and not for the general case I've written above).

In the literature I've found something in the book: Toro - Riemann solvers and numerical methods of fluid dynamics, 2009 but it is very short. Additionaly, I have find similar questions on Mathoverflow (question1 and question2)

Also,if anyone has a lot of time he/she could write this all down on a concrete example but the reference in the literature would be just fine.

In the literature it is easy to find books that show, step by step, how to get the solution of partial differential equation using various techniques, but it is not so easy to do the same for PDE systems.

Let's say we have first order, strictly hyperbolic, nonlinear, coupled 2x2 system of pdes, in one dimension, that looks like this: $$ \begin{cases} u_t+f(u,v) u_x=h(u,v) \\[2ex] v_t+g(u,v) v_x=m(u,v) \\[2ex] (u(x,0),v(x,0))=(u_0 (x), v_0 (x)) \end{cases} $$ (This is a Cauchy problem). If $(u_0 (x), v_0 (x))$ is discontinuous, e.g. $$(u(x,0),v(x,0))= \begin{cases} (u_l , v_l), x<0 \\[2ex] (u_r , v_r), x\geq0 \end{cases}$$ we would talk about Riemann problem.

I want to solve this problem using method of characteristics.

My question is: Is there any book/note/paper where I could find a step by step solution of quasilinear system that looks like this by using method of characteristics? Or maybe an easier quasilinear system (e.g. if we put right hand sides equal zero, put $f$ equal to $g$,...). It doesn't matter if it is a Riemann or Cauchy problem. I just want to see how system of this type could be solved in detail from start to finish. It could be shown on any concrete system (and not for the general case I've written above).

In the literature I've found something in the book:

Toro - Riemann solvers and numerical methods of fluid dynamics, 2009,

but it is very short. Additionaly, I have found a similar questions on Mathoverflow (question1 and question2).

If anyone has a lot of time he/she could write this all down on a concrete example but the reference in the literature would be just fine.

In the literature it is easy to find books that show, step by step, how to get the solution of partial differential equation, but it is not so easy to do the same for PDE systems.

Let's say we have first order, strictly hyperbolic, nonlinear, coupled 2x2 system of pdes, in one dimension, that looks like this: $$ \begin{cases} u_t+f(u,v) u_x=h(u,v) \\[2ex] v_t+g(u,v) v_x=m(u,v) \\[2ex] (u(x,0),v(x,0))=(u_0 (x), v_0 (x)) \end{cases} $$ (This is a Cauchy problem). If $(u_0 (x), v_0 (x))$ is discontinuous, e.g. $$(u(x,0),v(x,0))= \begin{cases} (u_l , v_l), x<0 \\[2ex] (u_r , v_r), x\geq0 \end{cases}$$ we would talk about Riemann problem.

I want to solve this problem using method of characteristics.

My question is: Is there any book/note/paper where I could find a step by step solution of system that looks like this by using method of characteristics? Or maybe an easier system (e.g. if we put right hand sides zero - in this case we wouldn't have sources). It doesn't matter if it is a Riemann or Cauchy problem.. I just want to see how this system could be solved in detail.

In the literature I've found something in the book: Toro - Riemann solvers and numerical methods of fluid dynamics, 2009 but it is very short. Additionaly, I have find similar questions on Mathoverflow (question1 and question2)

Also,if anyone has a lot of time he/she could write this all down on a concrete example but the reference in the literature would be just fine.

In the literature it is easy to find books that show, step by step, how to get the solution of partial differential equation, but it is not so easy to do the same for PDE systems.

Let's say we have first order, strictly hyperbolic, nonlinear, coupled 2x2 system of pdes, in one dimension, that looks like this: $$ \begin{cases} u_t+f(u,v) u_x=h(u,v) \\[2ex] v_t+g(u,v) v_x=m(u,v) \\[2ex] (u(x,0),v(x,0))=(u_0 (x), v_0 (x)) \end{cases} $$ (This is a Cauchy problem). If $(u_0 (x), v_0 (x))$ is discontinuous, e.g. $$(u(x,0),v(x,0))= \begin{cases} (u_l , v_l), x<0 \\[2ex] (u_r , v_r), x\geq0 \end{cases}$$ we would talk about Riemann problem.

I want to solve this problem using method of characteristics.

My question is: Is there any book/note/paper where I could find a step by step solution of quasilinear system that looks like this by using method of characteristics? Or maybe an easier quasilinear system (e.g. if we put right hand sides equal zero, put $f$ equal to $g$,...). It doesn't matter if it is a Riemann or Cauchy problem. I just want to see how system of this type could be solved in detail. It could be shown on any concrete system (and not for the general case I've written above).

In the literature I've found something in the book: Toro - Riemann solvers and numerical methods of fluid dynamics, 2009 but it is very short. Additionaly, I have find similar questions on Mathoverflow (question1 and question2)

Also,if anyone has a lot of time he/she could write this all down on a concrete example but the reference in the literature would be just fine.