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I assume, that you have conducted a space discretization, so that you are about solving the (vector-valued) ODE $$ \dot u_h(t) = F_h(t,u_h(t)), \text{ on [0,T] }, u_h(0) = \alpha. $$ via a numerical scheme $\Phi$ that advances the approximation $u_h^n$ at the current time instance $t=t^n$ to the next value $u_h^{n+1}$ at $t=t^{n+1}:=t^n+\tau$.

Then your questions refer to properties of explicit, where the update writes as $$ u_h^{n+1} = u_h^n + \Phi_e(t^n,\tau,u_h^n),\quad \quad \quad $$

implicit, written like $$ u_h^{n+1} = u_h^n + \Phi_i(t^n,\tau,u_h^{n+1},u_h^n), \quad \quad \quad (*) $$

or a combination of both ('IMEX', see e.g. here) single-step time-stepping schemes.

In this setup, the Newton method simply is an approach to solve the possibly nonlinear in $u_h^{n+1}$ systems resulting from $(*)$.

And my answers base on results from the numerical analysis of single-step methods.

1. If you use convergent schemes, in terms of the convergence order, there is no general advantage of using implicit schemes (see. 2.). However, for stiff systems, e.g. your system containing a Laplacian, there are implicit schemes that are stable without time-step restrictions. Nevertheless, in theory, for the explicit scheme, you get better results with smaller time-steps, as long as your equation itself is stable (e.g., referring to Picard-Lindelof Theorem, if $F_h$ is Lipshitz in the second argument) and your time-step is not too small.
2. You can find examples, where explicit schemes perform better. (Theoretically, you can reverse the time in your example, start from the terminal value, and find implicit and explicit interchanged.) If you make the Newton error sufficiently small, you can still improve accuracy by decreasing the time-step or by using time-stepping schemes of higher order.
3. The constant $C$ in the error estimate for the global error grows exponentially with the length of the time-interval. See, e.g., here for the explicit Euler scheme. This is true for every single-step method. As the estimate is of type $err \approx C \tau^p$, $p>0$, a smaller time-step $\tau$ only postpones this effect.

Some more remarks and the final answer:

• IMEX schemes can be used to treat only the linear part implicitly what avoids the nonlinear solves. See Jed Brown's answer.
• Crank-Nicolson is a single-step method. The 'multi' in multi-step methods refers to the use of a number of preceding timesteps to define the current update. E.g. like $$ u_h^{n+1} = \Phi_m(t^n,\tau,u_h^{n+1},u_h^n,u_h^{n-1}). $$ This is very different from single-step and also split-step or IMEX methods, where the update is defined not recurring to previous values.

So, my answer is: Yes, you can solve nonlinear PDEs without Newton's method. You can use explicit schemes, 'IMEX' schemes, or socalled linearly implicit methods (e.g. the Rosenbrock methods). Also, you can employ other approaches to solve the systems from $(*)$ like fixed-point iteration or, in particular cases, algebraic solvers.

I assume, that you have conducted a space discretization, so that you are about solving the (vector-valued) ODE $$ \dot u_h(t) = F_h(t,u_h(t)), \text{ on [0,T] }, u_h(0) = \alpha. $$ via a numerical scheme $\Phi$ that advances the approximation $u_h^n$ at the current time instance $t=t^n$ to the next value $u_h^{n+1}$ at $t=t^{n+1}:=t^n+\tau$.

Then your questions refer to properties of explicit, where the update writes as $$ u_h^{n+1} = u_h^n + \Phi_e(t^n,\tau,u_h^n),\quad \quad \quad $$

implicit, written like $$ u_h^{n+1} = u_h^n + \Phi_i(t^n,\tau,u_h^{n+1},u_h^n), \quad \quad \quad (*) $$

or a combination of both ('IMEX', see @Jed Brown's answer) single-step time-stepping schemes.

In this setup, the Newton method simply is an approach to solve the possibly nonlinear in $u_h^{n+1}$ systems resulting from $(*)$.

And my answers base on results from the numerical analysis of single-step methods.

1. If you use convergent schemes, in terms of the convergence order, there is no general advantage of using implicit schemes (see. 2.). However, for stiff systems, e.g. your system containing a Laplacian, there are implicit schemes that are stable without time-step restrictions. Nevertheless, in theory, for the explicit scheme, you get better results with smaller time-steps, as long as your equation itself is stable (e.g., referring to Picard-Lindelof Theorem, if $F_h$ is Lipshitz in the second argument) and your time-step is not too small.
2. You can find examples, where explicit schemes perform better. (Theoretically, you can reverse the time in your example, start from the terminal value, and find implicit and explicit interchanged.) If you make the Newton error sufficiently small, you can still improve accuracy by decreasing the time-step or by using time-stepping schemes of higher order.
3. The constant $C$ in the error estimate for the global error grows exponentially with the length of the time-interval. See, e.g., here for the explicit Euler scheme. This is true for every single-step method. As the estimate is of type $err \approx C \tau^p$, $p>0$, a smaller time-step $\tau$ only postpones this effect.

Some more remarks and the final answer:

• IMEX schemes can be used to treat only the linear part implicitly what avoids the nonlinear solves. See Jed Brown's answer.
• Crank-Nicolson is a single-step method. The 'multi' in multi-step methods refers to the use of a number of preceding timesteps to define the current update. E.g. like $$ u_h^{n+1} = \Phi_m(t^n,\tau,u_h^{n+1},u_h^n,u_h^{n-1}). $$ This is very different from single-step and also split-step or IMEX methods, where the update is defined not recurring to previous values.

So, my answer is: Yes, you can solve nonlinear PDEs without Newton's method. You can use explicit schemes, 'IMEX' schemes, or socalled linearly implicit methods (e.g. the Rosenbrock methods). Also, you can employ other approaches to solve the systems from $(*)$ like fixed-point iteration or, in particular cases, algebraic solvers.

I assume, that you have conducted a space discretization, so that you are about solving the (vector-valued) ODE $$ \dot u_h(t) = F_h(t,u_h(t)), \text{ on [0,T] }, u_h(0) = \alpha. $$ via a numerical scheme $\Phi$ that advances the approximation $u_h^n$ at the current time instance $t=t^n$ to the next value $u_h^{n+1}$ at $t=t^{n+1}:=t^n+\tau$.

Then your questions refer to properties of explicit, where the update writes as $$ u_h^{n+1} = u_h^n + \Phi_e(t^n,\tau,u_h^n),\quad \quad \quad $$

implicit, written like $$ u_h^{n+1} = u_h^n + \Phi_i(t^n,\tau,u_h^{n+1},u_h^n), \quad \quad \quad (*) $$

or a combination of both ('IMEX', see e.g. here) single-step time-stepping schemes.

In this setup, the Newton method is simply an approach to solve the possibly nonlinear in $u_h^{n+1}$ systems resulting from $(*)$.

And my answers base on results from the numerical analysis of single-step methods.

1. If you use convergent schemes, in terms of the convergence order, there is no general advantage of using implicit schemes (see. 2.). However, for stiff systems, e.g. your system containing a Laplacian, there are implicit schemes that are stable without time-step restrictions. Nevertheless, in theory, for the explicit scheme, you get better results with smaller time-steps, as long as your equation itself is stable (e.g., referring to Picard-Lindelof Theorem, if $F_h$ is Lipshitz in the second argument) and your time-step is not too small.
2. You can find examples, where explicit schemes perform better. (Theoretically, you can reverse the time in your example, start from the terminal value, and find implicit and explicit interchanged.) If you make the Newton error sufficiently small, you can still improve accuracy by decreasing the time-step or by using time-stepping schemes of higher order.
3. The constant $C$ in the error estimate for the global error grows exponentially with the length of the time-interval. See, e.g., here for the explicit Euler scheme. This is true for every single-step method. As the estimate is of type $err \approx C \tau^p$, $p>0$, a smaller time-step $\tau$ only postpones this effect.

Some more remarks and the final answer:

• IMEX schemes can be used to treat only the linear part implicitly what avoids the nonlinear solves. See Jed Brown's answer.
• Crank-Nicolson is a single-step method. The 'multi' in multi-step methods refers to the use of a number of preceding timesteps to define the current update. E.g. like $u_h^{k+1} = \Phi_m(t^n,\tau,u_h^{n+1},u_h^n,u_h^{n-1})$. This is very different from single-step and also split-step or IMEX methods, where the update is defined not recurring to previous values.

So, my answer is: Yes, you can solve nonlinear PDEs without Newton's method. You can use explicit schemes, 'IMEX' schemes, or socalled linear implicit methods (e.g. the Rosenbrock methods). Also, you can employ other approaches to solve the systems from $(*)$ like fixed-point iteration or, in particular cases, algebraic solvers.

I assume, that you have conducted a space discretization, so that you are about solving the (vector-valued) ODE $$ \dot u_h(t) = F_h(t,u_h(t)), \text{ on [0,T] }, u_h(0) = \alpha. $$ via a numerical scheme $\Phi$ that advances the approximation $u_h^n$ at the current time instance $t=t^n$ to the next value $u_h^{n+1}$ at $t=t^{n+1}:=t^n+\tau$.

Then your questions refer to properties of explicit, where the update writes as $$ u_h^{n+1} = u_h^n + \Phi_e(t^n,\tau,u_h^n),\quad \quad \quad $$

implicit, written like $$ u_h^{n+1} = u_h^n + \Phi_i(t^n,\tau,u_h^{n+1},u_h^n), \quad \quad \quad (*) $$

or a combination of both ('IMEX', see e.g. here) single-step time-stepping schemes.

In this setup, the Newton method simply is an approach to solve the possibly nonlinear in $u_h^{n+1}$ systems resulting from $(*)$.

And my answers base on results from the numerical analysis of single-step methods.

1. If you use convergent schemes, in terms of the convergence order, there is no general advantage of using implicit schemes (see. 2.). However, for stiff systems, e.g. your system containing a Laplacian, there are implicit schemes that are stable without time-step restrictions. Nevertheless, in theory, for the explicit scheme, you get better results with smaller time-steps, as long as your equation itself is stable (e.g., referring to Picard-Lindelof Theorem, if $F_h$ is Lipshitz in the second argument) and your time-step is not too small.
2. You can find examples, where explicit schemes perform better. (Theoretically, you can reverse the time in your example, start from the terminal value, and find implicit and explicit interchanged.) If you make the Newton error sufficiently small, you can still improve accuracy by decreasing the time-step or by using time-stepping schemes of higher order.
3. The constant $C$ in the error estimate for the global error grows exponentially with the length of the time-interval. See, e.g., here for the explicit Euler scheme. This is true for every single-step method. As the estimate is of type $err \approx C \tau^p$, $p>0$, a smaller time-step $\tau$ only postpones this effect.

Some more remarks and the final answer:

• IMEX schemes can be used to treat only the linear part implicitly what avoids the nonlinear solves. See Jed Brown's answer.
• Crank-Nicolson is a single-step method. The 'multi' in multi-step methods refers to the use of a number of preceding timesteps to define the current update. E.g. like $$ u_h^{n+1} = \Phi_m(t^n,\tau,u_h^{n+1},u_h^n,u_h^{n-1}). $$ This is very different from single-step and also split-step or IMEX methods, where the update is defined not recurring to previous values.

So, my answer is: Yes, you can solve nonlinear PDEs without Newton's method. You can use explicit schemes, 'IMEX' schemes, or socalled linearly implicit methods (e.g. the Rosenbrock methods). Also, you can employ other approaches to solve the systems from $(*)$ like fixed-point iteration or, in particular cases, algebraic solvers.

I assume, that you have conducted a space discretization, so that you are about solving the (vector-valued) ODE $$ \dot u_h(t) = F_h(t,u_h(t)), \text{ on [0,T] }, u_h(0) = \alpha. $$ Then your questions refer to properties of explicit, implicit (+ Newton to solve the implicit nonlinear equations), or both ('IMEX', see e.g. here) time-stepping schemes. And my answers base on results from the numerical analysis of single-step methods.

1. If you use convergent schemes, in terms of the convergence order, there is no general advantage of using implicit schemes (see. 2.). However, for stiff systems, e.g. your system containing a Laplacian, there are implicit schemes that are stable without time-step restrictions. Nevertheless, in theory, for the explicit scheme, you get better results with smaller time-steps, as long as your equation itself is stable (e.g., referring to Picard-Lindelof Theorem, if $F_h$ is Lipshitz in the second argument) and your time-step is not too small.
2. You can find examples, where explicit schemes perform better. (Theoretically, you can reverse the time in your example, start from the terminal value, and find implicit and explicit interchanged.) If you make the Newton error sufficiently small, you can still improve accuracy by decreasing the time-step or by using time-stepping schemes of higher order.
3. The constant $C$ in the error estimate for the global error grows exponentially with the length of the time-interval. See, e.g., here for the explicit Euler scheme. This is true for every single-step method. As the estimate is of type $err \approx C \tau^p$, $p>0$, a smaller time-step $\tau$ only postpones this effect.

I assume, that you have conducted a space discretization, so that you are about solving the (vector-valued) ODE $$ \dot u_h(t) = F_h(t,u_h(t)), \text{ on [0,T] }, u_h(0) = \alpha. $$ via a numerical scheme $\Phi$ that advances the approximation $u_h^n$ at the current time instance $t=t^n$ to the next value $u_h^{n+1}$ at $t=t^{n+1}:=t^n+\tau$.

Then your questions refer to properties of explicit, where the update writes as $$ u_h^{n+1} = u_h^n + \Phi_e(t^n,\tau,u_h^n),\quad \quad \quad $$

implicit, written like $$ u_h^{n+1} = u_h^n + \Phi_i(t^n,\tau,u_h^{n+1},u_h^n), \quad \quad \quad (*) $$

or a combination of both ('IMEX', see e.g. here) single-step time-stepping schemes.

In this setup, the Newton method is simply an approach to solve the possibly nonlinear in $u_h^{n+1}$ systems resulting from $(*)$.

And my answers base on results from the numerical analysis of single-step methods.

1. If you use convergent schemes, in terms of the convergence order, there is no general advantage of using implicit schemes (see. 2.). However, for stiff systems, e.g. your system containing a Laplacian, there are implicit schemes that are stable without time-step restrictions. Nevertheless, in theory, for the explicit scheme, you get better results with smaller time-steps, as long as your equation itself is stable (e.g., referring to Picard-Lindelof Theorem, if $F_h$ is Lipshitz in the second argument) and your time-step is not too small.
2. You can find examples, where explicit schemes perform better. (Theoretically, you can reverse the time in your example, start from the terminal value, and find implicit and explicit interchanged.) If you make the Newton error sufficiently small, you can still improve accuracy by decreasing the time-step or by using time-stepping schemes of higher order.
3. The constant $C$ in the error estimate for the global error grows exponentially with the length of the time-interval. See, e.g., here for the explicit Euler scheme. This is true for every single-step method. As the estimate is of type $err \approx C \tau^p$, $p>0$, a smaller time-step $\tau$ only postpones this effect.

Some more remarks and the final answer:

• IMEX schemes can be used to treat only the linear part implicitly what avoids the nonlinear solves. See Jed Brown's answer.
• Crank-Nicolson is a single-step method. The 'multi' in multi-step methods refers to the use of a number of preceding timesteps to define the current update. E.g. like $u_h^{k+1} = \Phi_m(t^n,\tau,u_h^{n+1},u_h^n,u_h^{n-1})$. This is very different from single-step and also split-step or IMEX methods, where the update is defined not recurring to previous values.

So, my answer is: Yes, you can solve nonlinear PDEs without Newton's method. You can use explicit schemes, 'IMEX' schemes, or socalled linear implicit methods (e.g. the Rosenbrock methods). Also, you can employ other approaches to solve the systems from $(*)$ like fixed-point iteration or, in particular cases, algebraic solvers.